Arsene Wenger's side had come in for criticism for surrendering a three-goal advantage to draw 3-3 with Anderlecht in the UEFA Champions League on Tuesday and they were at it again on Sunday.
In-form forward Alexis Sanchez's seventh goal in his last six games looked to have put the visitors on course for a third consecutive victory in the top flight, but Swansea struck twice in the space of three minutes to secure the points.
Victory for Swansea was only their second in eight Premier League outings and moved them above Arsenal into fifth, with Wenger's men slipping a place further back heading into the international break.
Tottenham loanee Tom Carroll replaced the suspended Jonjo Shelvey in Swansea's midfield with the returning Leon Britton on the bench to enhance Garry Monk's options, while Kyle Bartley and Marvin Emnes replaced the injured Federico Fernandez and Wayne Routledge.
Jefferson Montero's pace was causing right-back Chambers problems and the Ecuador winger forced a save from Szczesny at his near post and the Arsenal goalkeeper was called into action again to deny Emnes after Bony had picked the former Middlesbrough man out.
Lukasz Fabianski was finally called into action a minute before the break against his former club to save Danny Welbeck's strike after the England international had played a one-two with Alex Oxlade-Chamberlain as Arsenal finished the first half strongly.
Oxlade-Chamberlain had a chance of his own just after the break when Alexis played him in, but the midfielder got the ball stuck under his feet before his scuffed shot was easily saved by Fabianski, who was up to the test to deny Santi Cazorla's shot that skidded off the wet surface as rain lashed down.
And worse was to come for Arsenal, with Gomis rising to head home Montero's cross soon after replaced Bony to stun the visitors.
