Nyasha Mushekwi continued his form in the absence of want-away striker Katlego Mphela, scoring an early goal that set the tone, while defender Method Mwanjali made it 2-0 midway through the second half.
The result is yet another blow to the Birds, who remain second from bottom on the log and next have a tough away game against Soweto rivals Kaizer Chiefs.
The Brazilians got off to a perfect start, with Mushekwi scoring in the first minute of the game with a low shot.
There were no further notable attacking moments in the first half, though Swallows certainly looked desperate to try and get a foothold in the game but could find no way through the visitors’ defence.
The Birds shuffled their pack at the start of the second half, sending on striker Brad Ritson in place of the ineffective Shaun Haschick, but the Birds still lacked a potent goal threat.
Sundowns almost doubled their lead in the 53rd minute, with Elias Pelembe firing a good shot on target from a free kick, but Greg Etafia made a fine save to keep his side in contention.
Yet there was nothing the Nigerian shot-stopper could do to prevent Mwanjali’s goal in the 68th minute, with the Zimbabwean defender crashing home a shot from the edge of the box to kill off the game.
Sundowns took their foot off the pedal after the second goal, and even a debut for marquee signing Teko Modise didn’t really spark them back to life, but they had done enough to wrap up the points.
The win sees Sundowns move to third place on 35 points, leapfrogging Ajax Cape Town who drew away to Bloemfontein Celtic.
Swallows are still in 15th place, three points ahead of Black Aces but a full six points adrift of the safety of 14th place, presently occupied by Maritzburg United.