Leon Osman's deft second-half finish rescued a point for Everton as they finally broke Norwich's resistance.
The hosts dominated throughout and created a series of chances, but it was City who took the lead when Grant Holt turned and shot low into the corner.
Louis Saha came closest to levelling before the break with a fizzing low effort and the chances kept coming.
The pressure finally told when Osman diverted in Royston Drenthe's strike to deny the Canaries a rare away victory.
Substitute Drenthe then almost won it with a fierce, swerving strike that overworked visiting goalkeeper John Ruddy did well to block with his shoulder.
Before Drenthe's late cameo, Norwich looked set to secure their second away Premier League victory this season thanks to a combination of resolute defending and sloppy home finishing.
But Everton's determination and refusal to accept a fifth home defeat in eight league games this season finally told.
As commendable as Norwich's display was, Everton could easily have wrapped up the game by the time Holt had fired the Canaries ahead.
Osman, Marouane Fellaini and Phil Neville got to grips with the midfield and allowed Everton to dominate possession and territory from the outset.
Fellaini had the first chance after a delightful first-time pass from Tim Cahill, but the big-haired Belgian's poked effort was brilliantly blocked by Ruddy.
Osman then cut in from the left and shot towards goal, but the effort was headed away to safety by the backpedalling Russell Martin.
Magaye Gueye also volleyed wide after Ruddy flapped at a clearance following a series of corners and Cahill sent a tame effort wide.
A home goal seemed inevitable, but when the deadlock was broken it was the visitors celebrating.
Canaries forward Steve Morison knocked David Fox's floated free-kick back into the danger area and Holt showed great strength, composure and awareness to swivel neatly in the six-yard box and shoot into the net via the inside of the far post.
Holt's contribution - his seventh goal in 14 league games this season - could have ended soon after when he appeared to elbow Fellaini in the jaw.
The Belgian clumsily challenged Holt and seemed to catch him around the face. The burly striker reacted before instantly protesting his innocence.
Referee Lee Probert decided not to book either, but had he got a good view of the incident, Holt could easily have been dismissed.
With Everton rattled, Wes Hoolahan squandered a great chance to make it 2-0 when he profited from a Neville slip, but overhit his cross.
The same pattern of relentless home pressure continued after the interval, with Saha the biggest threat.
DID YOU KNOW?
Norwich's last top-flight away win over Everton was also their biggest Premier League victory on their travels. Efan Ekoku scored four goals as the Canaries coasted to a 5-1 success.
Osman had a shot turned around the post by Ruddy after a great run and unselfish pass from Saha and the Frenchman then forced a brilliant instinctive block from Ruddy following Denis Stracqualursi's flick-on.
Martin also blocked Saha's shot from the edge of the box after a rare, rash Ruddy moment saw the keeper come racing out of his goal.
At the other end, Norwich's Holt almost headed a second.
But the Goodison Park gloom was lifted when Drenthe's low strike was cleverly flicked in by fit-again midfielder Osman.
The home pressure increased even more after the leveller. Saha went close again and Drenthe twice had efforts saved by Ruddy.
Marc Tierney's stunning tackle also denied Stracqualursi as Everton were forced to settle from a point - only their eighth from eight home games this season.
Everton manager David Moyes:
"I'm delighted we got a point in the end but disappointed we didn't win. If we had scored a goal in the first 20 minutes I think we would have gone on and got the win.
"We started very well had some good opportunities but didn't take them.
"Overall we played okay. We lost our way a little bit before half-time but we kept going and we deserved a win.
"But a draw was more than I was thinking I was going to get today towards the end."
Norwich City manager Paul Lambert:
"This is a really hard place to come and we battled really strongly and defended really strongly when we had to but also looked a threat.
"They came rolling right at us and you weather the storm, that is what you have to do and that is what we did at Anfield.
"Any point away from home is good. Holt has been great for the last two-and-a-half years I have been here. I think he has scored 50-odd goals in 100 or so games and in anyone's book that is a phenomenal ratio.
"He has always been a handful and it is a great testament to him he is probably getting the reward from it. It was a brilliant finish."
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Live Text Commentary
Full Time The final whistle is blown by the referee.90:00 +2:06
Leon Osman challenges David Fox unfairly and gives away a free kick. John Ruddy restarts play with the free kick.90:00 +1:25
Foul by Grant Holt on Marouane Fellaini, free kick awarded. Johnny Heitinga restarts play with the free kick.90:00 +0:53
Royston Drenthe produces a left-footed shot from just outside the area that goes harmlessly over the target.89:51
Conor McAleny takes a shot. John Ruddy makes a save.88:52
Shot by Conor McAleny from deep inside the penalty area misses to the left of the goal.87:50 Substitution
Substitution Louis Saha leaves the field to be replaced by Conor McAleny.87:27
Tony Hibbert takes a shot. Save by John Ruddy.86:20
Corner taken by Leon Osman from the left by-line to the near post, Marc Tierney manages to make a clearance.86:02
Royston Drenthe takes a shot. Save by John Ruddy.84:19
Corner taken left-footed by Royston Drenthe from the right by-line to the near post, clearance made by Marc Tierney.83:37
Headed effort from the edge of the area by Bradley Johnson goes wide of the right-hand upright.82:42
Royston Drenthe takes a shot. John Ruddy makes a save.80:34
The assist for the goal came from Royston Drenthe.80:34 Goal scored
Goal! - Leon Osman - Everton 1 - 1 Norwich Leon Osman finds the net with a goal from inside the penalty box to the bottom right corner of the goal. Everton 1-1 Norwich.80:33
Royston Drenthe takes a shot. Blocked by Leon Osman.79:38
Free kick awarded for an unfair challenge on Marc Tierney by Royston Drenthe. John Ruddy restarts play with the free kick.78:49 Substitution
Substitution Aaron Wilbraham is brought on as a substitute for Steve Morison.77:36
Kyle Naughton produces a right-footed shot from outside the penalty box and misses right.77:01
The free kick is delivered right-footed by David Fox from left channel, clearance by Zak Whitbread.
Booking The referee shows Phil Neville a yellow card.76:51
Free kick awarded for a foul by Phil Neville on Bradley Johnson.75:34
Phil Neville fouled by Andrew Crofts, the ref awards a free kick. Free kick taken by Phil Neville.74:02
Foul by Johnny Heitinga on Grant Holt, free kick awarded. Kyle Naughton takes the free kick.73:05 Substitution
Substitution Bradley Johnson on for Wesley Hoolahan.72:54
Header by Louis Saha from deep inside the penalty area misses to the left of the target.71:29
Free kick awarded for an unfair challenge on Phil Jagielka by Grant Holt. Free kick taken by Phil Neville.70:55 Substitution
Substitution Royston Drenthe on for Magaye Gueye.70:46
Close range headed effort by Louis Saha goes wide of the left-hand upright.69:27
Unfair challenge on Denis Stracqualursi by Zak Whitbread results in a free kick. Phil Neville takes the free kick.68:13
Denis Stracqualursi concedes a free kick for a foul on David Fox. Free kick taken by Kyle Naughton.65:21
Magaye Gueye takes a corner.62:35
Louis Saha takes a shot. Blocked by Russell Martin.61:46
Inswinging corner taken by Leon Osman from the left by-line, Zak Whitbread makes a clearance.61:30
Inswinging corner taken by Leighton Baines, clearance by Russell Martin.61:17
Louis Saha takes a shot. Save made by John Ruddy.60:42 Substitution
Substitution Denis Stracqualursi is brought on as a substitute for Tim Cahill.60:42
The official flags Grant Holt offside. Johnny Heitinga restarts play with the free kick.59:48
Magaye Gueye takes a shot.58:58
Foul by Andrew Crofts on Louis Saha, free kick awarded. Free kick taken by Leighton Baines.58:06
Leon Osman has an effort at goal from outside the penalty area which goes wide right of the goal.57:15
Grant Holt has a headed effort at goal from deep inside the area missing to the right of the goal.56:34
Headed effort from inside the area by Tim Cahill misses to the right of the goal.54:57
Louis Saha takes a shot. Russell Martin gets a block in.54:36
Grant Holt is adjudged to have handled the ball. Free kick taken by Johnny Heitinga.53:30
Corner taken by Leon Osman from the left by-line, Close range headed effort by Johnny Heitinga misses to the right of the goal.53:17
Leon Osman takes a shot. Save made by John Ruddy.52:26
Grant Holt is flagged offside by the assistant referee. Indirect free kick taken by Johnny Heitinga.51:41
The official flags Leighton Baines offside. Indirect free kick taken by John Ruddy.48:18
Inswinging corner taken left-footed by Magaye Gueye from the right by-line, save by John Ruddy.45:01
The game restarts for the second half.45:00 +2:08 Half time
Half Time The whistle is blown to end the first half.45:00 +0:48
Unfair challenge on Grant Holt by Phil Neville results in a free kick. John Ruddy restarts play with the free kick.44:01
Tim Cahill is adjudged to have handled the ball. John Ruddy takes the indirect free kick.43:40
Close range header by Grant Holt misses to the right of the target.42:54
Tim Cahill concedes a free kick for a foul on Marc Tierney. The free kick is swung in left-footed by Andrew Surman.38:43
Unfair challenge on Tim Cahill by Zak Whitbread results in a free kick. Direct strike on goal from the free kick comes in from Leighton Baines, Kyle Naughton gets a block in.38:16
Marc Tierney concedes a free kick for a foul on Louis Saha. Tony Hibbert takes the indirect free kick.37:02
Marouane Fellaini is flagged offside by the assistant referee. Indirect free kick taken by Russell Martin.35:49
Grant Holt concedes a free kick for a foul on Johnny Heitinga. Johnny Heitinga restarts play with the free kick.32:13
Louis Saha produces a right-footed shot from just outside the area that goes wide left of the goal.31:29
Louis Saha takes a shot. Marc Tierney gets a block in.27:46
The assist for the goal came from Steve Morison.27:46 Goal scored
Goal! - Grant Holt - Everton 0 - 1 Norwich Grant Holt fires in a goal from inside the six-yard box to the bottom right corner of the goal. Everton 0-1 Norwich.27:16
Johnny Heitinga concedes a free kick for a foul on Wesley Hoolahan. David Fox restarts play with the free kick.26:20
Andrew Surman challenges Leon Osman unfairly and gives away a free kick. Indirect free kick taken by Tim Howard.22:47
Tim Cahill takes a shot. Marc Tierney gets a block in.19:33
Headed effort from inside the area by Leon Osman misses to the left of the goal.18:58
Tim Cahill takes a shot.18:33
Johnny Heitinga takes a shot. Kyle Naughton gets a block in.17:49
Andrew Surman is penalised for a handball. Leighton Baines takes the free kick.17:14
Leon Osman takes a inswinging corner from the left by-line played to the near post, Marc Tierney makes a clearance.15:54
Magaye Gueye takes a inswinging corner from the right by-line to the near post.14:16
Unfair challenge on David Fox by Marouane Fellaini results in a free kick. Andrew Surman takes the free kick.13:37
Magaye Gueye has an effort at goal from outside the area which goes wide left of the target.13:16
Leon Osman takes a inswinging corner from the left by-line to the near post, John Ruddy makes a clearance.12:00
Corner taken by Magaye Gueye from the right by-line, Zak Whitbread makes a clearance.11:30
Leon Osman takes the inswinging corner, John Ruddy makes a save.11:09
Leon Osman takes a shot. Russell Martin gets a block in.8:49
Inswinging corner taken right-footed by Leon Osman from the left by-line, Zak Whitbread makes a clearance.8:16
Grant Holt produces a volleyed right-footed shot from 12 yards. Save made by Tim Howard.7:02
Handball by Marouane Fellaini. Indirect free kick taken by John Ruddy.6:18
Marouane Fellaini takes a shot. Save by John Ruddy.5:44
Free kick awarded for a foul by Marc Tierney on Leon Osman. Tony Hibbert takes the free kick.4:25
Steve Morison takes a long range shot harmlessly clearing the crossbar.3:33
Johnny Heitinga restarts play with the free kick.3:33 Booking
Booking The referee shows Kyle Naughton a yellow card for unsporting behaviour.3:17
Foul by Kyle Naughton on Leighton Baines, free kick awarded.2:12
Grant Holt gives away a free kick for an unfair challenge on Leighton Baines. Tim Howard takes the free kick.0:00
The game begins.
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