The Gunners had by far the better of the contest and Nacho Monreal's first goal for the club gave them the lead with 16 minutes remaining, before substitute Gervinho struck in stoppage time.
The win moves Arsene Wenger's side to within two points of fourth-placed Chelsea - and four of third-placed Tottenham - with their rivals in action on Sunday.
Arsenal could have led in the fifth minute thanks to Alex Oxlade-Chamberlain as he tricked his way past Angel Rangel to beat Michel Vorm, only for his shot to clip the top of the bar.
Swansea quickly settled after the early scare, and Ki Sung-yueng's weighted pass gave Rangel a run in on goal, but the full-back sliced wide.
Santi Cazorla's dipping volley was parried by Vorm but the Swans were looking assured, and the Gunners' high defensive line were relieved to see Michu flagged offside after being cut apart by another superb through ball from Ki.
Oxlade-Chamberlain was again denied by the woodwork three minutes before the break, his fierce swerving strike had Vorm beat all ends up, but flicked the crossbar as it flew over.
Swansea's attacking forays were becoming increasingly rare and the dam eventually broke in the 74th minute.
Cazorla was again heavily involved as he tricked and teased his way past defenders on the edge of the box. The chance looked to have gone when Olivier Giroud took a heavy touch, but Monreal pounced to lash a shot across Vorm and into the corner with the help of a deflection off Ashley Williams.
Gervinho then made the points safe after Aaron Ramsey's pass teed him up to slot low beyond Vorm.