After a tight first half the away side showed its class, with Serge Gnabry and Aaron Ramsey producing top finishes, while Ben Davies' goal proved just a consolation
Ramsey's hot streak of form – which has seen him score five goals in his last four matches – continued in his native Wales as Arsene Wenger's men won their fifth Premier League game in a row.
After an uninspiring first half, Ramsey set up German youngster Serge Gnabry for the game's first goal in the 58th minute.
And the former Cardiff City midfielder produced a composed finish himself just a few moments later as Arsenal recorded its eighth Premier League away win in succession.
With a lengthy injury list, Arsenal – which is missing the likes of Santi Cazorla, Theo Walcott, Lukas Podolski and Alex Oxlade-Chamberlain – made nine changes from the side that beat West Brom on penalties in the League Cup on Wednesday, retaining only German pair Gnabry and Per Mertesacker.
Swansea saw its defense of the League Cup end at Birmingham City in midweek, and it kept just four players in the team, recalling Spanish star Michu, although the Swans were without injured defensive pair Ashley Williams and Angel Rangel (both ankle).
Despite both sides enjoying good spells of possession, they failed to penetrate in a dull first half that saw neither produce a shot on target.
The visitors should have scored in first-half stoppage time though, with an excellent Gnabry run finishing with Olivier Giroud, but the Frenchman, who had gone down with an ankle injury a few moments earlier, dragged his effort wide.
The second half started slowly, with Szczesny forced to charge off his line to deny Michu, but Arsenal won the game thanks to four minutes of brilliance.
The lively Gnabry opened the scoring just short of the hour, making the most of a neat Ramsey flick to slot a cool finish beyond Swansea goalkeeper Michel Vorm and into the bottom corner.
Mesut Ozil nearly doubled Arsenal's lead less than a minute later but saw his near-post effort saved by Vorm, before a counterattack ended with Ramsey continuing his scoring run.
An excellent team move that also involved Jack Wilshere and Giroud saw the latter cut back for Ramsey, and the goalscorer took a touch before smashing an effort into the top corner to delight the traveling fans.
Nathan Dyer forced Szczesny into a terrific save as Swansea pressed for a goal, and defender Ben Davies gave his side a lifeline with eight minutes to play.
The left back ran forward and played in substitute Wilfried Bony, and Davies followed his run, collecting Bony's lobbed pass and finishing beyond Szczesny.
It was a terrific goal but time ran out for Swansea, as Arsenal won again ahead of a mouth-watering UEFA Champions League clash against Rafael Benitez's Napoli on Tuesday.