The win moved Arsene Wenger's side to within two points of fourth-place Chelsea, 24 hours before Rafael Benitez's side face West Ham United at Stamford Bridge.
The Gunners struggled to break down Michael Laudrup's side but eventually made the breakthrough when Monreal scored his first goal since arriving at the club in an £8.3 million ($13.2 million, 9.7 million euros) move from Malaga in January.
Then, with Swansea looking for a leveller, Gervinho secured the win in added time. Wenger had called on his side to display the qualities they had shown in beating Bayern Munich in mid-week, when they went out of the Champions League on away goals.
The message appeared to get through, with Arsenal pressing forward immediately and only narrowly failing to match their efforts in Munich, when Olivier Giroud had set them on their way with a third-minute goal.
This time the early opportunity fell to Alex Oxlade-Chamberlain, who tricked his way past Swansea right-back Angel Rangel before firing a right-foot shot that rebounded to safety off the crossbar.
But Swansea were equally positive in their response and Rangel made his presence felt at the opposite end of the pitch when he broke clear in the visiting area, only to slice his shot wide.
Swansea's victory at Newcastle United two weeks previously had taken the League Cup winners to the 40-point target set by Laudrup at the start of the season, prompting him to set a new target of a top-half finish.
His side looked capable of taking the first step towards that mark when Michu turned Per Mertesacker in the 17th minute, only to pull his left-foot shot across the face of Lukasz Fabianski's goal.
Fabianski had again been preferred to Wojciech Szczcesny in the Arsenal goal, with Szczesny taking his place on the substitutes' bench alongside Thomas Vermaelen, the club captain.
In Vermaelen's absence, the centre-back pairing of Laurent Koscielny and Mertesacker had impressed in Munich and the duo again provided a solid barrier in front of Fabianski, denying Swansea for the rest of the first half.
And with more luck, the visitors might have reached the interval ahead, as Oxlade-Chamberlain again hit the bar with another long-range effort after good work from the impressive Santi Cazorla three minutes before the break.
Arsenal maintained their momentum after the restart, with Abou Diaby firing over from outside the Swansea box.
The home side had looked like a team sat comfortably in mid-table, but they began to show more urgency as the second half progressed, with Michu and Pablo Hernandez firing over from good positions.
But it was Arsenal who finally made the breakthrough, three minutes after Wenger had freshened up his side with the introduction of Aaron Ramsey and Gervinho for Oxlade-Chamberlain and Diaby.
A period of sustained pressure brought about a corner that Swansea failed to clear effectively.
The home defence was then stretched by a strong Cazorla run across the face of goal and when the home side again failed to clear the Spaniard's low cross, Monreal was on hand to finish left-footed.
Swansea pressed for an equaliser but were caught out on the break when Ramsey sped down the right before squaring for Gervinho to finish from 12 yards.