Steve Morison's late equaliser earned Norwich a draw that left a major dent in Arsenal's Champions League hopes.
Yossi Benayoun curled in a shot with barely a minute gone but Wojciech Szczesny somehow allowed Wes Hoolahan's tame effort to beat him.
Grant Holt's deflected shot put Norwich ahead but Robin van Persie levelled with a volley and completed the comeback with a well-hit second.
But Morison's cross-shot pulled the Canaries level in a thrilling finish.
The result extends Arsenal's winless run to four games, and seriously damages their chances of qualifying for the Champions League.
Arsene Wenger's fate is no longer in his own hands - the Gunners, who are two points clear in third place, can be overhauled on Sunday when north London rivals Tottenham travel to Aston Villa and Newcastle host Manchester City.
Did you know?
Benayoun became the first Arsenal player to score a Premier League goal on his birthday since Robert Pires on 29 October 2005, at Tottenham (1-1).
After last week's
1-1 draw at Stoke
, Wenger admitted that his team were over-reliant on Van Persie's goals, and this performance did little to disprove that view.
Van Persie, who only has one year remaining on his contract, thought he had once again come to his side's rescue when he extended his remarkable goal tally to 37, but Norwich substitute Morison had other ideas.
Wenger could point to referee Anthony Taylor's failure to award a penalty for Kyle Naughton's blatant push on Van Persie in stoppage time but Norwich's positive approach and impressive late defending certainly merited a point.
The Arsenal manager had insisted that there should be no need for nerves from his players, but even if there was any anxiety it should have been disappeared when Benayoun swept in.
Remaining fixtures in the race for third and fourth
6 May - Aston Villa v
6 May -
v Manchester City
13 May - Everton v
13 May -
13 May - West Brom v
Tomas Rosicky seized on a loose clearance to pick out the Israeli, who used an overlapping run from Kieran Gibbs to dummy and create the space to bend a shot into the far corner.
Despite their recent slump, Norwich responded well and they pulled level when Naughton pulled back and Szczesny made a mess of Hoolahan's first-time shot.
The visitors pulled ahead just before the half-hour, when Hoolahan threaded through for Holt to hit a shot that looped up off Gibbs and over Szczesny.
After losing Bacary Sagna to what appeared to be an ankle problem, things could have got even worse for the hosts.
Arsenal's miserable run
16 April -
Arsenal 1 Wigan 2
21 April -
Arsenal 0 Chelsea 0
28 April -
Stoke 1 Arsenal 1
5 May -
Arsenal 3 Norwich 3
Benayoun got away with kicking out at Ryan Bennett and Norwich had valid penalty appeals for Francis Coquelin's sliding tackle on Hoolahan and Laurent Koscielny's tug on Martin.
Simeon Jackson should have added a third after the break when he tore away from Thomas Vermaelen only to fail to trouble Szczesny.
Rosicky's free-kick picked out Benayoun but the on-loan Chelsea midfielder headed straight at John Ruddy.
It was by no means one-way traffic, though, and Hoolahan clipped a volley wide from a neat Holt flick at the other end.
Van Persie had barely been involved but the Dutchman made no mistake after 71 minutes when he volleyed in from Alex Song's fine pass.
Lambert praises 'brilliant' Norwich
The Dutch international had a stabbed effort held by Ruddy but put Arsenal in front when his shot from just inside the area had enough power to beat Ruddy.
But Morison pulled the visitors level once again when he raced on to a through ball and drove a shot across Szczesny for his first goal since January.
Still the chances came thick and fast. Ruddy saved from Van Persie, who was then denied a late penalty when he went down under a barge from Naughton before the Norwich goalkeeper pulled off a fine block from Song.
And Ruddy was outstanding once again as he somehow got a hand to a header from Gibbs, giving Norwich a well-deserved point which could have far-reaching consequences.
Arsenal manager Arsene Wenger:
"Here the whole season we have got zero penalties. It is absolutely amazing, because you see other clubs who have got 10.
"I don't know (why the penalty was not given), we have to accept the decision of the referee.
"It is more than frustrating because we were very poor in the first half, not switched on."
Norwich manager Paul Lambert:
"I've said to the players that I'm so proud of what they've given this season and this highlighted it even more. We were outstanding.
"I thought Spurs was as good as anything in my time here but this might surpass it.
"We wanted to get on the front foot and everyone was outstanding. We could have been 3-1 up when Simeon went through and should have had a penalty in the first half. They might think they should have had a penalty at the end but Gervinho was offside."
Full Time The game is over, as the referee blows his whistle.
Steve Morison concedes a free kick for a foul on Laurent Koscielny. Direct free kick taken by Laurent Koscielny.
Kieran Gibbs takes a shot. Save by John Ruddy.
Alex Oxlade-Chamberlain takes a shot. Anthony Pilkington gets a block in. Corner taken by Robin van Persie from the right by-line, Alex Oxlade-Chamberlain takes a shot. Save by John Ruddy.
Wojciech Szczesny takes the direct free kick.
90:00 +1:58 Booking
Booking Aaron Wilbraham goes into the book for unsporting behaviour.
Free kick awarded for a foul by Aaron Wilbraham on Laurent Koscielny.
Alex Song takes a shot. John Ruddy makes a save.
Header by Marouane Chamakh from deep inside the penalty area misses to the right of the target.
Robin van Persie takes a shot. Save made by John Ruddy.
The assist for the goal came from Jonathan Howson.
84:29 Goal scored
Goal - Steve Morison - Arsenal 3 - 3 Norwich Steve Morison finds the back of the net with a goal from inside the area to the bottom left corner of the goal. Arsenal 3-3 Norwich.
Shot from just outside the area by Bradley Johnson goes over the bar.
Substitution (Norwich) makes a substitution, with Aaron Wilbraham coming on for Grant Holt.
79:03 Goal scored
Goal - Robin van Persie - Arsenal 3 - 2 Norwich A goal is scored by Robin van Persie from just inside the penalty area high into the middle of the goal. Arsenal 3-2 Norwich.
A cross is delivered by Kieran Gibbs, John Ruddy makes a comfortable save.
The referee blows for offside against Steve Morison. Wojciech Szczesny takes the free kick.
Thomas Vermaelen takes a shot. Blocked by Ryan Bennett. Tomas Rosicky takes a shot. Bradley Johnson gets a block in. Inswinging corner taken by Tomas Rosicky from the left by-line, clearance by Russell Martin.
Grant Holt gives away a free kick for an unfair challenge on Alex Song. Thomas Vermaelen takes the direct free kick.
Robin van Persie takes a shot. John Ruddy makes a comfortable save.
The offside flag is raised against Steve Morison. Alex Song takes the free kick.
Substitution Anthony Pilkington replaces Wesley Hoolahan.
Inswinging corner taken by Robin van Persie from the right by-line, Marouane Chamakh has a headed effort from close range and clears the bar.
Alex Song provided the assist for the goal.
71:16 Goal scored
Goal - Robin van Persie - Arsenal 2 - 2 Norwich Robin van Persie finds the back of the net with a goal from inside the six-yard box to the bottom left corner of the goal. Arsenal 2-2 Norwich.
Marouane Chamakh takes a shot. Blocked by Russell Martin.
Thomas Vermaelen fouled by Grant Holt, the ref awards a free kick. Wojciech Szczesny takes the direct free kick.
Free kick taken by John Ruddy.
Substitution Marouane Chamakh joins the action as a substitute, replacing Yossi Benayoun.
Substitution Simeon Jackson goes off and Steve Morison comes on.
Free kick awarded for a foul by Tomas Rosicky on Simon Lappin.
Corner taken by Robin van Persie from the right by-line, clearance made by Grant Holt.
Thomas Vermaelen takes a shot. Blocked by Ryan Bennett. Tomas Rosicky takes a shot. Bradley Johnson gets a block in. Inswinging corner taken by Tomas Rosicky, Russell Martin makes a clearance.
Effort on goal by Robin van Persie from inside the penalty box misses to the right of the target.
Wesley Hoolahan concedes a free kick for a foul on Tomas Rosicky. The free kick is delivered left-footed by Robin van Persie from right channel, clearance by Grant Holt.
Substitution Alex Oxlade-Chamberlain replaces Aaron Ramsey.
Foul by Jonathan Howson on Yossi Benayoun, free kick awarded. The free kick is delivered right-footed by Tomas Rosicky from left channel, Jonathan Howson makes a clearance.
Kieran Gibbs restarts play with the free kick.
Booking Wesley Hoolahan is booked.
Kieran Gibbs fouled by Wesley Hoolahan, the ref awards a free kick.
Robin van Persie takes a shot. Save by John Ruddy. Aaron Ramsey has an effort at goal from deep inside the area missing to the left of the goal.
Wesley Hoolahan produces a right-footed shot from deep inside the penalty box which goes wide of the left-hand upright.
Free kick awarded for an unfair challenge on Alex Song by Elliott Bennett. A cross is delivered by Tomas Rosicky, Yossi Benayoun takes a shot. John Ruddy makes a comfortable save.
Laurent Koscielny challenges Grant Holt unfairly and gives away a free kick. Jonathan Howson restarts play with the free kick.
Simeon Jackson takes a shot. Save by Wojciech Szczesny.
Centre by Robin van Persie.
Tomas Rosicky takes a inswinging corner from the left by-line to the near post, clearance by Kyle Naughton.
Simeon Jackson takes a shot. Blocked by Laurent Koscielny.
The second half has started.
45:00 +4:20 Half time
Half Time The whistle is blown to end the first half.
Headed effort from deep inside the penalty area by Grant Holt misses to the left of the goal.
Ryan Bennett takes the direct free kick.
45:00 +2:59 Booking
Booking Caution for Thomas Vermaelen.
Thomas Vermaelen gives away a free kick for an unfair challenge on Simeon Jackson.
Foul by Grant Holt on Francis Coquelin, free kick awarded. Direct free kick taken by Francis Coquelin.
Grant Holt fouled by Laurent Koscielny, the ref awards a free kick. Free kick crossed by Elliott Bennett, clearance by Laurent Koscielny. Outswinging corner taken by Elliott Bennett from the right by-line played to the near post, Francis Coquelin makes a clearance.
The assistant referee signals for offside against Grant Holt. Alex Song takes the free kick. Bradley Johnson challenges Tomas Rosicky unfairly and gives away a free kick. Robin van Persie crosses the ball from the free kick left-footed from right channel, clearance made by Ryan Bennett.
Simon Lappin fouled by Yossi Benayoun, the ref awards a free kick. John Ruddy takes the direct free kick.
Effort from inside the six-yard box by Kieran Gibbs goes wide right of the goal.
Strike from close range by Simeon Jackson clears the bar.
Free kick awarded for an unfair challenge on Francis Coquelin by Wesley Hoolahan. Francis Coquelin takes the free kick.
Substitution Francis Coquelin comes on in place of Bacary Sagna.
Booking Simeon Jackson receives a caution.
Free kick awarded for a foul by Grant Holt on Laurent Koscielny. Free kick taken by Laurent Koscielny.
Assist on the goal came from Wesley Hoolahan.
26:06 Goal scored
Goal - Grant Holt - Arsenal 1 - 2 Norwich Goal scored by Grant Holt from just inside the area low into the middle of the goal. Arsenal 1-2 Norwich.
Simon Lappin concedes a free kick for a foul on Bacary Sagna. Bacary Sagna takes the direct free kick.
Free kick awarded for a foul by Bradley Johnson on Bacary Sagna. Bacary Sagna takes the free kick.
Elliott Bennett takes the free kick.
Booking Aaron Ramsey goes into the referee's book for unsporting behaviour.
Foul by Aaron Ramsey on Jonathan Howson, free kick awarded.
The assistant referee signals for offside against Robin van Persie. Free kick taken by John Ruddy.
Grant Holt is flagged offside by the assistant referee. Wojciech Szczesny takes the indirect free kick.
Inswinging corner taken by Tomas Rosicky from the left by-line played to the near post, clearance by Simon Lappin.
Effort from the edge of the area by Robin van Persie goes wide of the left-hand upright.
Simon Lappin challenges Yao Gervinho unfairly and gives away a free kick. Bacary Sagna takes the free kick.
Kyle Naughton provided the assist for the goal.
11:56 Goal scored
Goal - Wesley Hoolahan - Arsenal 1 - 1 Norwich Wesley Hoolahan finds the back of the net with a goal from inside the area to the bottom right corner of the goal. Arsenal 1-1 Norwich.
Header from close range by Grant Holt clears the bar.
Free kick awarded for a foul by Yao Gervinho on Simon Lappin. Bradley Johnson delivers the ball from the free kick left-footed from left channel. The official flags Grant Holt offside. Free kick taken by Wojciech Szczesny.
Elliott Bennett has an effort at goal from outside the area missing to the wide left of the goal.
Grant Holt takes a shot. Kieran Gibbs gets a block in.
Bradley Johnson fouled by Tomas Rosicky, the ref awards a free kick. Free kick taken by Russell Martin.
Assist by Tomas Rosicky.
1:05 Goal scored
Goal - Yossi Benayoun - Arsenal 1 - 0 Norwich Yossi Benayoun scores a goal from the edge of the penalty area to the top right corner of the goal. Arsenal 1-0 Norwich.
The ref blows the whistle to start the match.
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