Bafetimbi Gomis struck with five minutes to play as the Swans recorded their fifth league double of the season with victory at the Emirates.
The visitors frustrated the home side in the first half with the Gunners restricted to a string of half chances.
The hosts dominated for long spells in the second period but former Arsenal goalkeeper Lukasz Fabianski was in inspired form as he made a string of fine stops - including a spectacular double save to deny Alexis Sanchez and Theo Walcott.
But substitute Gomis had the decisive say as he headed a late winner - which was awarded after goal-line technology proved it had crossed the line - to propel the Swans to within a point of seventh-placed Southampton.
Garry Monk made just one change to the starting line-up as Ki Sung-Yueng - who came off the bench to score in the win over Stoke - replaced the injured Nelson Oliveira.
Lukasz Fabianski made his first visit to the Emirates Stadium since leaving the Gunners last summer while Bafe Gomis was included on the bench for the first time since recovering from a hamstring problem.
David Ospina, Hector Bellerin, Per Mertesacker (capt), Laurent Koscielny, Nacho Monreal, Francis Coquelin, Santi Cazorla, Aaron Ramsey, Mesut Ozil, Alexis Sanchez, Olivier Giroud.
Wojciech Szczesny, Kieran Gibbs, Gabriel, Mathieu Flamini, Jack Wilshere, Tomas Rosicky, Theo Walcott.
Lukasz Fabianski, Angel Rangel, Federico Fernandez, Ashley Williams (capt), Neil Taylor, Jack Cork, Ki Sung-Yueng, Jonjo Shelvey, Gylfi Sigurdsson, Nathan Dyer, Jefferson Montero.
Gerhard Tremmel, Jazz Richards, Leon Britton, Matt Grimes, Modou Barrow, Marvin Emnes, Bafe Gomis.
The Swans came into this fixture on the back of wins over Newcastle and Stoke; results which had consolidated their grip on eighth place in the Barclays Premier League.
Arsenal, meanwhile, were unbeaten since February and had lost just once in their last 34 home leagues.
The Gunners had only conceded twice in their last seven outings and the started on the front foot as Olivier Giroud saw an early header land on the roof of the net after neat build-up around the penalty area.
Swansea were looking a threat on the counter attack and Angel Rangel had a low shot comfortably held by David Ospina at the other end.
Giroud failed to test Lukasz Fabianski with a left-foot effort from 25 yards, while the lively Jefferson Montero failed to hit the target when he shot wide with the final kick of the half.
Half-time: Arsenal 0 Swans 0
Neither side made any changes at the break, but the game sprung into the life in the early stages as Ki Sung-Yueng saw a shot deflect wide and Federico Fernandez produced a fine saving challenge to deny the onrushing Giroud inside the six-yard box.
Alexis Sanchez then shot into the side-netting as the Gunners probed for an opener; with Giroud involved again just shy of the hour mark as his shot on the turn was well held by Fabianski.
The visitors were forced into their first change with half an hour to play as Angel Rangel was replaced by Jazz Richards, while his Wales international team-mate Aaron Ramsey could only find the side netting after a blistering counter attack from the hosts, and Fabianski was again called into action on 65 minutes as he kept out Santi Cazorla's strike from just inside the penalty area.
The Gunners were upping the tempo by now and made a double change as Jack Wilshere and Theo Walcott were introduced in place of Francis Coquelin and Olivier Giroud.
And Walcott was involved almost immediately as he saw a tame effort held by Fabianski before the Swans made their second change as the returning Bafe Gomis replaced Ki Sung-Yueng.
Fabianski - on his first return to the Emirates - was now playing a busy role as he saved from both Nacho Monreal and Aaron Ramsey in quick succession.
But the best was yet to come from the Poland international as he produced a superb double save to deny Sanchez and Walcott from 12 yards out.
Swansea made a third and final change with ten minutes to play as Modou Barrow replaced Nathan Dyer.
And the substitute played a key role as Garry Monk's men stunned the Emirates by breaking the deadlock with five minutes remaining.
The winger clipped a cross to the far edge of the box where the ball found its way to Montero, who then delivered for Bafe Gomis to rise highest to plant a firm header goalwards that crossed the line after a moment of hesitation around the stadium, with referee Kevin Friend signalling the goal.
That then sparked a furious late rally from the Gunners as they sought an equaliser - but they found their former goalkeeper in inspired form on his return to north London and the points headed straight back to SA1 as the Swans completed their sixth league double of the season to move within a point of seventh-placed Southampton.