After several near misses in the first half, the second half looked to be petering out into a stalemate until Monreal’s scuffed effort beat Michel Vorm with 17 minutes remaining, and Gervinho added a second in the final minutes.
The victory takes Arsenal onto 50 points, just two points behind Chelsea, albeit having played one game more.
Swansea had outplayed Arsenal en route to claiming three points at the Emirates earlier this season and they again began brightly on their own turf.
The home side showed off the attractive passing style that has won them so many plaudits, with one particularly impressive move leading to Sung-Yueng Ki playing a fine pass into the forward running Angel Rangel but, under pressure, the full back sliced his shot wide of the post.
As well as their passing, Swansea’s high pressing has also been a feature of their game and that was also in full evidence. Michael Laudrup’s side seized on a poor pass from Carl Jenkinson on the edge of the box but top scorer Michu dragged his shot just past the post.
Arsenal were struggling to mount any concerted pressure, with their only serious threats to the opposition goal in the first half coming courtesy of two pieces of individual skill from Alex Oxlade-Chamberlain. First the midfielder, playing in place of Lukas Podolski, turned his defender inside and out on the left of the box before curling a shot toward the far corner that struck flush on the cross bar.
Late in the half, Oxlade Chamberlain fired a rasping effort from 25 yards that this time bounced off the top of the cross bar with Michel Vorm well beaten in the Swansea goal.
There second half was one of few opportunities, with Pablo Hernandez having the only meaningful efforts on goal early in the period, but Arsenal beginning to assert themselves more as the period wore on. With 16 minutes remaining Arsenal struck a vital opening goal.
The chance looked to be gone when Olivier Giroud took the ball away from a better placed opponent with a poor touch following a cross, but the ball came back to Monreal and the January signing from Malaga struck a scuffed shot that had just on it to beat the dive of Vorm.
Having gone in front, Arsenal’s defense again featuring Laurent Koscielny alongside Per Mertesacker and in front of Lukasz Fabianksi were holding out with little fuss.
And with Swansea committing men upfield, Arsenal struck a second on the counter. Giroud released Aaron Ramsey in the clear and the Welshman laid the ball across to fellow-substitute Gervinho to tap in.