A late equaliser from Swansea striker Danny Graham capped a frantic spell of three goals in six minutes and booked the Welsh club an FA Cup replay with Arsenal.
The Gunners' eight-year wait for a trophy seemed set to continue as Michu put Swansea ahead two minutes after coming off the bench.
But Lukas Podolski, another substitute, levelled with nine minutes remaining and left-back Kieran Gibbs crashed in a spectacular volley two minutes later.
Arsenal's joy was short-lived, however, as Graham was left unmarked at the far post to lift the ball into the roof of the net and keep the hosts in with a chance of a fourth-round trip to Brighton.
- Arsenal have not failed to make the fourth round of the FA Cup since 1996
- In three previous FA Cup meetings between the sides Arsenal have won two and Swansea one.
- Arsenal have won the FA Cup 10 times, Swansea have never reached the final.
There was still time for both sides to create chances to win a match that started slowly but ended in a frantic exchange of goalmouth action.
Sixth in the Premier League, out of the Capital One Cup and drawn against Bayern Munich in the Champions League last 16, Arsenal were desperate to keep their most realistic chance of silverware alive.
But when Michu came off the bench to fire Swansea ahead just before the hour, another disappointment looked on the cards.
As Nathan Dyer's header bounced just inside the Arsenal half, Michu was the quickest to react and lifted the ball over Gunners defender Per Mertesacker.
That gave him a clear run at Wojciech Szczesny and Michu showed characteristic calmness to slot past the goalkeeper with a side-foot finish reminiscent of his two goals in Swansea's 2-0 Premier League win at the Emirates Stadium in December.
Michu's goal - his 15th in 24 appearances this season - came just as Arsenal were enjoying their best moments of the match.
Olivier Giroud twice forced saves from Michel Vorm with driven efforts from inside the area.
In between those shots, Aaron Ramsey screamed for a penalty when he fell as he passed Dwight Tiendalli, although replays suggested the Welshman tripped over his own feet.
After a toothless first-half display that hardly troubled the home side, Arsenal suddenly found some urgency and directness.
And like Michu, substitute Podolski wasted little time in making his presence felt after replacing Ramsey.
The German showed a striker's instinct to swivel and shoot in one smooth motion and find the bottom corner.
Two minutes later, the tie swung in Arsenal's favour as Gibbs exchanged passes with Mikel Arteta and crashed in a volley from a tight angle.
But when Kemy Agustien's corner fell to the feet of Sung-Yeung Ki, the South Korea international showed exemplary composure to set up Graham, who was left in far too much space at the far post.
The striker took a touch and lifted the ball over a diving Szczesny.
There was still drama to come in the closing minutes as Laurent Koscielny's header was palmed away by home goalkeeper Vorm. The rebound fell perfectly for the Arsenal defender but he fired wildly over the bar.
And in stoppage time, Swansea right-back Tiendalli found himself in space inside Arsenal's box but drove a low effort into the side netting.
The replay will take place at Emirates Stadium on Wednesday 16 January.
Live Text Commentary
Full Time The final whistle is blown by the referee.90:00 +4:28
Unfair challenge on Mikel Arteta by Sung-Yeung Ki results in a free kick. Theo Walcott delivers the ball from the free kick right-footed from right wing, save by Michel Vorm.90:00 +2:57
The ball is sent over by Dwight Tiendalli. The ball is delivered by Ben Davies, clearance made by Mikel Arteta.90:00 +2:41
A cross is delivered by Dwight Tiendalli, clearance by Kieran Gibbs.90:00 +1:32
The ball is delivered by Santi Cazorla, save by Michel Vorm.90:00 +1:18
The ball is crossed by Santi Cazorla, Kyle Bartley makes a clearance.90:00 +0:31
Dwight Tiendalli produces a right-footed shot from deep inside the penalty area which goes wide of the right-hand post.88:51
Corner taken by Theo Walcott, Laurent Koscielny takes a shot. Save made by Michel Vorm. Laurent Koscielny takes a shot from close range clearing the crossbar.88:16
Centre by Ben Davies, save made by Wojciech Szczesny.87:54
Per Mertesacker challenges Danny Graham unfairly and gives away a free kick. Chico takes the free kick.87:19
Kieran Gibbs fouled by Sung-Yeung Ki, the ref awards a free kick. Direct free kick taken by Mikel Arteta.86:08
Assist on the goal came from Sung-Yeung Ki.86:08 Goal scored
Goal! - Danny Graham - Swansea 2 - 2 Arsenal Danny Graham grabs a goal from close range to the top right corner of the goal. Swansea 2-2 Arsenal.85:55
Corner taken by Kemy Agustien.85:19
Sung-Yeung Ki takes a shot. Save by Wojciech Szczesny. Nathan Dyer takes a shot. Laurent Koscielny gets a block in.84:58
The referee gives a free kick against Mikel Arteta for handball. Kemy Agustien restarts play with the free kick.83:17 Booking
Booking The referee shows Kieran Gibbs a yellow card.82:48
The assist for the goal came from Olivier Giroud.82:48 Goal scored
Goal! - Kieran Gibbs - Swansea 1 - 2 Arsenal Kieran Gibbs finds the net with a goal from close in to the top left corner of the goal. Swansea 1-2 Arsenal.82:24
Kieran Gibbs produces a cross, Ben Davies makes a clearance.
Assist on the goal came from Mikel Arteta.80:46 Goal scored
Goal! - Lukas Podolski - Swansea 1 - 1 Arsenal Lukas Podolski fires in a goal from inside the penalty area to the bottom left corner of the goal. Swansea 1-1 Arsenal.80:25
Inswinging corner taken from the left by-line by Theo Walcott.80:17
A cross is delivered by Kieran Gibbs, clearance by Kyle Bartley.79:04
Santi Cazorla shoots direct from the free kick.79:04 Booking
Booking Kyle Bartley is cautioned.78:54
Jack Wilshere fouled by Kyle Bartley, the ref awards a free kick.76:53
The assistant referee flags for offside against Lukas Podolski. Michel Vorm takes the free kick.76:15
Kieran Gibbs crosses the ball, Kyle Bartley gets a block in.75:47
Unfair challenge on Laurent Koscielny by Miguel Michu results in a free kick. Direct free kick taken by Mikel Arteta.74:14
Theo Walcott produces a right-footed shot from deep inside the penalty box which goes wide of the left-hand upright.73:34
Unfair challenge on Chico by Santi Cazorla results in a free kick. Direct free kick taken by Chico.72:52
Bacary Sagna sends in a cross, Ben Davies gets a block in.71:51 Substitution
Substitution Lukas Podolski is brought on as a substitute for Aaron Ramsey.68:09 Substitution
Substitution Kemy Agustien comes on in place of Leon Britton.67:49
Jack Wilshere crosses the ball, save by Michel Vorm.66:16
Ben Davies gives away a free kick for an unfair challenge on Bacary Sagna. Mikel Arteta takes the free kick.65:35
Jack Wilshere produces a left-footed shot from just outside the box that misses to the right of the net.65:22
The referee blows for offside. Laurent Koscielny restarts play with the free kick.65:00
The ball is swung over by Aaron Ramsey, Olivier Giroud has a headed effort at goal from deep inside the area missing to the right of the goal.63:57
Leon Britton fouled by Jack Wilshere, the ref awards a free kick. Kyle Bartley takes the direct free kick.63:27
Pablo Hernandez delivers the ball, save by Wojciech Szczesny.62:02
Corner taken by Theo Walcott, clearance by Miguel Michu.61:03
Bacary Sagna concedes a free kick for a foul on Ben Davies. Free kick crossed by Sung-Yeung Ki, clearance made by Olivier Giroud.60:27
Jack Wilshere concedes a free kick for a foul on Sung-Yeung Ki. Direct free kick taken by Kyle Bartley.59:54
Dwight Tiendalli is caught offside. Wojciech Szczesny takes the indirect free kick.57:14
Assist on the goal came from Danny Graham.57:14 Goal scored
Goal! - Miguel Michu - Swansea 1 - 0 Arsenal A goal is scored by Miguel Michu from just outside the box to the bottom left corner of the goal. Swansea 1-0 Arsenal.56:28
A cross is delivered by Kieran Gibbs.55:24 Substitution
Substitution Miguel Michu on for Jonathan de Guzman.55:24 Substitution
Substitution (Swansea) makes a substitution, with Pablo Hernandez coming on for Wayne Routledge.53:11
Kieran Gibbs delivers the ball, Olivier Giroud takes a shot. Michel Vorm makes a save.52:10
Inswinging corner taken right-footed by Theo Walcott from the left by-line, save by Michel Vorm.50:38
Drilled left-footed shot by Olivier Giroud. Save made by Michel Vorm.50:22
Ben Davies produces a cross, Kieran Gibbs makes a clearance.49:18
Aaron Ramsey delivers the ball, clearance made by Kyle Bartley.47:38
Olivier Giroud has an effort at goal from just outside the box which goes wide of the right-hand post.46:58
Centre by Kieran Gibbs, save by Michel Vorm.45:27
Danny Graham is flagged offside by the assistant referee. Laurent Koscielny takes the free kick.45:01
The second half begins.45:00 +0:23 Half time
Half Time The half-time whistle blows.43:30
Unfair challenge on Kyle Bartley by Olivier Giroud results in a free kick. Direct free kick taken by Kyle Bartley.43:02
Santi Cazorla takes a shot. Save made by Michel Vorm.41:43
The ball is swung over by Aaron Ramsey, Michel Vorm makes a save.40:38
Free kick awarded for an unfair challenge on Jack Wilshere by Sung-Yeung Ki. Free kick taken by Mikel Arteta.40:23
Wayne Routledge produces a cross, Laurent Koscielny manages to make a clearance.39:06
Jonathan de Guzman delivers the ball, Per Mertesacker manages to make a clearance.38:05
The ball is delivered by Ben Davies, clearance by Kieran Gibbs.37:54
The ball is delivered by Dwight Tiendalli, clearance by Laurent Koscielny.37:09
Nathan Dyer gives away a free kick for an unfair challenge on Kieran Gibbs. Free kick taken by Mikel Arteta.34:55
Jonathan de Guzman crosses the ball, Kyle Bartley takes a shot.34:55 Booking
Booking Aaron Ramsey booked for unsporting behaviour.34:49
Aaron Ramsey concedes a free kick for a foul on Dwight Tiendalli.34:08
Santi Cazorla produces a left-footed shot from inside the area that goes over the bar.33:42
Free kick awarded for an unfair challenge on Leon Britton by Olivier Giroud. Chico takes the direct free kick.32:47
Ben Davies delivers the ball.30:28
Dwight Tiendalli sends in a cross, Wojciech Szczesny makes a save.27:14
Foul by Aaron Ramsey on Dwight Tiendalli, free kick awarded. Kyle Bartley takes the free kick.26:29
The ball is swung over by Mikel Arteta.25:28
The ball is delivered by Kieran Gibbs, Ben Davies makes a clearance.21:28
Chico challenges Theo Walcott unfairly and gives away a free kick. Shot on goal comes in from Theo Walcott from the free kick.19:54
Theo Walcott takes a shot. Blocked by Kyle Bartley.18:42
Inswinging corner taken by Jonathan de Guzman, clearance by Bacary Sagna.17:04
Foul by Wayne Routledge on Bacary Sagna, free kick awarded. Wojciech Szczesny restarts play with the free kick.15:36
Jonathan de Guzman has an effort at goal from outside the box which goes wide right of the target.14:16
Free kick awarded for an unfair challenge on Theo Walcott by Chico. Jack Wilshere crosses the ball from the free kick left-footed from right wing, save made by Michel Vorm.13:11
Foul by Jonathan de Guzman on Kieran Gibbs, free kick awarded. Kieran Gibbs restarts play with the free kick.12:47
Danny Graham takes a shot. Save made by Wojciech Szczesny.11:16
Ben Davies concedes a free kick for a foul on Theo Walcott. Free kick crossed by Jack Wilshere, save by Michel Vorm.10:22
Theo Walcott delivers the ball.9:48
Danny Graham is ruled offside. Free kick taken by Per Mertesacker.9:10
The ball is sent over by Dwight Tiendalli, Bacary Sagna makes a clearance.6:58
The referee blows for offside against Danny Graham. Wojciech Szczesny takes the indirect free kick.6:38
The referee gives a free kick against Kieran Gibbs for handball. Free kick taken by Dwight Tiendalli.5:11
Nathan Dyer is flagged offside by the assistant referee. Wojciech Szczesny takes the indirect free kick.4:22
Free kick awarded for a foul by Aaron Ramsey on Dwight Tiendalli. Direct free kick taken by Dwight Tiendalli.4:05
Unfair challenge on Nathan Dyer by Jack Wilshere results in a free kick. Sung-Yeung Ki takes the direct free kick.3:42
Wayne Routledge sends in a cross, Wojciech Szczesny makes a comfortable save.3:03
Inswinging corner taken by Jack Wilshere, clearance by Kyle Bartley.2:15
Jack Wilshere fouled by Sung-Yeung Ki, the ref awards a free kick. Jack Wilshere crosses the ball from the free kick left-footed from right channel, clearance made by Chico.0:00
The match begins.
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