Wenger stuck with the same side that thrashed Everton 5-2 last weekend and they eventually came up with the goods to secure a 2-1 win and move within a point of rivals Tottenham in third place.
Sam Clucas had given Swansea a first-half lead with his maiden goal for the club but Sead Kolasinac levelled after the interval before laying on Aaron Ramsey's winner seven minutes later.
While Wenger stuck with his big-name attacking trio of Mesut Ozil, Alexandre Lacazette and Alexis Sanchez, it was Kolasinac - as well as midfield pairing Ramsey and Granit Xhaka - who shone.
Paul Clement set up defensively at the Emirates Stadium as the Swans looked for their fourth league win in the last six years here.
Arsenal set to their task immediately but failed to break down a determined defensive unit in the opening exchanges.
And the visitors made them pay as they opened the scoring in simple style, Clucas latching on to Tammy Abraham's pass to ghost into the box unmarked and finish low past Petr Cech.
The game slipped back into the same routine but still Arsenal could not create any chances, with Clucas firing wide as Swansea again looked to hit them on the break.
It took until five minutes before the interval for the Gunners to work their former goalkeeper but Lukasz Fabianski produced a superb stop to prevent Sanchez levelling.
Swansea were almost gifted a two-goal half-time lead as Per Mertesacker was caught in possession inside his own box, with Jordan Ayew stealing the ball from the Arsenal skipper but opting to shoot rather than square to Tammy Abraham, who was free in front of goal.
It took less than six minutes of the second half for Arsenal to level, Lacazette feeding Ozil whose shot was deflected out for Kolasinac to arrow home.
Swansea's fans thought they had retaken the lead immediately but Abraham was flagged offside before finishing high into the net.
Laurent Koscielny missed a glorious chance to turn the game around as the France defender could only send a free header wide from Sanchez's free-kick.
It did not take long for the second Arsenal goal to come, however, Kolasinac collecting a cross-field pass from Granit Xhaka before crossing in for Ramsey to sweep home.
Like for much of the contest, Arsenal then controlled proceedings without being able to add to their lead, substitute Olivier Giroud coming closest as he bent an effort inches wide of Fabianski's post in the dying moments.