Michael Laudrup insisted the first goal was the difference as Arsenal inflicted a rare home defeat on his side.
Second half goals from Nacho Monreal and substitute Gervinho proved the difference as Arsene Wenger’s men claimed all three points at the Liberty Stadium.
The home side had two good chances in the first half as Michu and Angel Rangel spurned good opportunities, while Alex Oxlade-Chamberlain hit the crossbar twice for the visitors.
It was only Swansea’s third home defeat of the season, but Laudrup’s men remain in the top half of the table with just eight games left to play.
“In the first half there was some good football from both sides,” said the Dane. “They had a couple of chances, but so did we.
“In the second half there wasn’t a great deal of football from either side, but the first goal decided the game in my opinion.
“We knew it was a massive game for Arsenal and they needed a win today.”
The Swans now have the luxury of a two-week break due to international fixtures before hosting in-form Tottenham at the Liberty at the end of March.
And with Manchester City, Chelsea and Manchester United still to come in the remaining eight games, Laudrup insisted that this players need to maintain their focus.
“It’s always important to try and find motivation in every game,” he said. “Everyone wants to win and the attitude today was fine.
“It isn’t very often that you see a small team like us sitting with 40 points with eight games to go and a major trophy in our hands.”
Meanwhile, the result means Arsenal stay fifth in the Premier League table – just two points adrift of fourth-placed Chelsea.
And boss Arsene Wenger saluted his players after they claimed a crucial three points in SA1 – their first win at the Liberty since the Swans won promotion to the top flight.
“Our attitude was outstanding,” he said. “We knew before the game what was expected and I’d like to congratulate the players on the win.
“We’ll only know the importance of this win at the end of the season but we knew we couldn’t afford to drop points.
“It’s always difficult to win in the Champions League and then win away in the Premier League a few days after.”