Siyabonga Nomvete scored a penalty in the first half, after a foul by Method Mwanjali, who was sent off.
Swallows narrowly held on and were lucky to come away with all three points, despite having an extra man for most of the match.
Sundowns started the game brightly, with Anthony Laffor splitting the Swallows defence as soon as the fifth minute. Teko Modise played the Liberian into space but he failed to find the target with only the keeper to beat.
Sundowns dominated the early exchanges, with Laffor and especially Modise running the show in midfield. Swallows seemed to have forgotten how to play in the first 15 minutes of the game, with the Brazilians having 80 percent of the possession.
The Birds opened the scoring against the run of play when Nomvete netted his fifth goal of the season from the penalty spot, after David Mathebula was brought down from behind in the box by Mwanjali, who received a straight red.
The second half started much like the first with the Brazilians pressing forward, although Lerato Chabangu went close for the visitors with a fierce shot from the edge of the area on the hour mark, hitting the side netting.
Nyasha Mushekwi almost found the equaliser in the 73rd minute, when he got on the end of a Katlego Mphela chip, but his volley went narrowly wide of Greg Etafia’s post.
Mushekwi made a clear impact off the bench and again went close to restoring parity in the 82nd minute, his header from close range going just over the bar.