The Gunners started the weekend five points adrift of fourth-placed Chelsea - seven behind third-placed Tottenham - and could not afford to drop any more points in the race for Champions League qualification.
Second-half goals from Nacho Monreal - his first for the club - and substitute Gervinho ensured they remain firmly in the mix, with their rivals both in action on Sunday.
Alex Oxlade-Chamberlain had struck the bar twice in the first half, before Monreal swept home with 16 minutes remaining after Olivier Giroud's poor touch fell into his path.
Gervinho then made sure of the points in stoppage time with a low finish from Aaron Ramsey's pass.
And Wenger was delighted with the spirit of his players as they became just the third side to win at the Liberty Stadium this season.
He said: "We will know at the end of the season how important this win is, but we knew as well the win was absolutely vital for us and we could not drop points today [Saturday].
"We have shown we are up for the fight and we will do so until the end of the season.
"The attitude of the players was outstanding, even when our legs were less than fresh after playing in midweek.
"We kept going and refused to drop points. The players knew what was expected.
"What we have done in the last three days is very difficult to do.
"To win away in the Champions League and then win away in the Premier League is very difficult and I would like to congratulate the players for that."
Under Wenger, Arsenal have long been admired for their style in possession, but it was a rock-solid defensive display that laid the foundations for victory, with Swansea barely testing keeper Lukasz Fabianski.
The Frenchman was delighted with the solidity of his side, for whom Abou Diaby was magnificent in midfield.
Wenger said: "They defended well, as did we. It was a game of defensive stability and we just needed an opening and the first goal was vital.
"We defended high up the pitch and in the whole second half we tried to pin them back, but we did not create many clear chances until the last 20 minutes and once we went 1-0 up I thought more would come."
Swansea had threatened during a bright start, with Angel Rangel and Michu failing to take presentable chances, before Arsenal took a grip on the game.
But manager Michael Laudrup felt a draw would have been a fair result.
He said: "I think in the first half we saw some good football from both sides.
"Our philosophies are similar and they had a couple of chances with Chamberlain hitting the bar, we had some good chances too.
"In the second half there was less good football. There was a lot of play in midfield but no shots on target.
"I thought it would be 0-0, but when things are like that the first goal will decide the game and that is what happened, it was a strange goal, but it was a goal and that decided the game."