The visitors took the lead through Sam Clucas, but goals from Sead Kolasinac and former Cardiff City midfielder Aaron Ramsey secured a 10th successive league win for the Gunners at the Emirates Stadium.
Swansea had their moments in North London, and Paul Clement will argue that Tammy Abraham should not have had a goal disallowed for offside moments after Arsenal's equaliser, but the visitors have now only recorded four points from seven matches in the top flight.
The hosts continued to control the pace of the match, with Sanchez again going close with a curling effort from inside the box which missed the target, but completely against the run of play, Swansea went in front.
Chelsea loanee Abraham threaded a terrific ball through to the advancing Clucas, who kept his composure to send the ball through the legs of Petr Cech for his first goal since a summer move from Hull City.
Arsenal offered very little in response to going behind against a team who had only conceded one goal in their last five away games in the top flight, although Sanchez was providing a threat each time he got on the ball in and around the penalty area.
As the game approached half time, the forward received the ball in the box, but Fabianski was equal to his rising effort, which appeared destined for the roof of the net.
Just before the break, Jordan Ayew should have doubled Swansea's lead but despite only having Cech to beat, the Ghanaian international was superbly denied as he attempted to flick the ball over the goalkeeper.
Arsenal needed to improve after the restart and it took the home side just six minutes to get back on level terms through Kolasinac, who received the ball at the back post from Mesut Ozil and proceeded to lash a shot into the far corner of Fabianski's net from 14 yards.
Moments later, Abraham had the ball in the net for Swansea but was harshly adjudged to have been in an offside position, and Arsenal soon set about capitalising on their bit of good fortune.
Laurent Koscielny powered a header marginally wide of the post but two minutes later, Arsenal were in front through Ramsey who directed the ball into the corner via Fabianski after collecting Kolasinac's pass.
Swansea were struggling to make much of an impact in the final third but Abraham did go close after turning on the edge of the area, only to see his deflected strike go wide of Cech's goal.
That proved to be the most significant action of the closing minutes as Arsenal moved to within a point of rivals Tottenham Hotspur, who sit in third.