The Spanish left-back beat Michel Vorm late on and Gervinho added a second at the Liberty Stadium to leave the Gunners trailing fourth-place Chelsea by only two points
The new signing from Malaga pounced on a loose ball in the penalty area after 74 minutes before beating Michel Vorm with a low shot that brings Arsene Wenger's team within two points of Chelsea, who play West Ham on Sunday.
Alex Oxlade-Chamberlain twice hit the crossbar but both goalkeepers had remained largely untested until Monreal struck and Gervinho slotted home in added time on the counterattack.
The left-back replaced Kieran Gibbs in the Arsenal XI, with Tomas Rosicky dropping out for Alex Oxlade-Chamberlain and Santi Cazorla moving to a more central position. Thomas Vermaelen was once again restricted to a place among the substitutes.
Oxlade-Chamberlain stepped past Angel Rangel and hit the crossbar with a sidefooted shot from 12 yards but the home side soon took control, stifling Arsenal's attacks with the numbers they brought back behind the ball.
A neat move in midfield ended in Ki Sung-Yueng threading a through ball into the path of Rangel, but under pressure from two Arsenal defenders the right-back miscued his shot.
Michu did much the same when Carl Jenkinson's poor pass across the edge of his own penalty area was intercepted by Jonathan de Guzman. The Spaniard checked back to wriggle away from Per Mertesacker but scuffed his effort from 15 yards wide of the far post.
Thereafter, though, Swansea struggled to work clear sights of goal just as much as the visitors. Oxlade-Chamberlain again clipped the crossbar, this time from 30 yards, before the half was out but the tempo was desperately slow after the restart.
Michu and Pablo Hernandez both missed the target with speculative efforts and with the two Spaniards, as well as Theo Walcott, fairly quiet throughout, it was difficult to see where a match-winning spark would come from.
Arsenal, though, would find the breakthrough from an unlikely source. Olivier Giroud looked to have wasted a good chance when he controlled a Cazorla cross poorly in the Swansea box, but the loose ball ran to Monreal, who drilled a low shot past Vorm.
Laudrup responded by introducing Moore but Wenger's side went close again when substitute Gervinho cut inside only to fire straight at the goalkeeper.
Swansea's captain, Ashley Williams, had already reminded his team-mates that they must not become complacent after their Capital One Cup victory but De Guzman's free kick against the side netting was the closest they came to an equaliser as they slipped to a second straight defeat.
They were always likely to be prone to an Arsenal counterattack as they poured forward in search of a late goal and Gervinho took advantage by finishing off a break led by Aaron Ramsey.