The Welsh outfit came into the clash at St Mary's Stadium without a victory in four Premier League outings, but ended that sequence as Jonjo Shelvey's stunning late goal saw them take all three points, while the hosts saw Ryan Bertrand sent off late on for a dangerous challenge on Modou Barrow.
Southampton dominated possession for much of the encounter, and went close through James Ward-Prowse and Sadio Mane.
However, a fine save from Lukasz Fabianski denied the former and Ashley Williams pulled off a double goal-line clearance to keep out Mane as Swansea sealed an impressive triumph.
"Tactically, I think we got it spot on," said Monk.
"We frustrated them in the first half, then we went for the three points in the second. And what a goal by Jonjo Shelvey - that's a goal worthy of winning any game.
"I'm happy with the three points in what has been a difficult month for us. We spoke about getting back to basics in the week, and we executed the plan very well.
"We've been in a difficult moment, especially in the last two games.
"We didn't show the commitment we've shown all season. But today was a very committed performance.
"There are periods in a season when you have to adapt. We've played some fantastic football this season, but you have to adapt sometimes, and we did that very well today."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.