The former Bafana Bafana mentor, leading the Brazilians in a caretaker capacity following the dismissal of Gordon Igesund on Sunday, saw his side fall to a seventh home defeat in the 2007/08 season despite taking the lead just a few minutes into the game.
Sundowns signalled their intentions early on when Peter Ndlovu's shot forced Swallows 'keeper Greg Etafia into a good save in the first minute.
And less than two minutes later the Brazilians opened the scoring, with Brent Carelse latching onto a loose ball in the penalty area, firing into the back of the net after Lerato Chabangu's shot had been parried by Etafia.
Swallows, though, did not allow the concession of an early goal to put them off and they drew level in the ninth minute, with Zambian striker James Chamanga hitting a low drive past Brian Baloyi.
The Birds, buoyed by their equaliser, went on the attack, but Mark Haskins, Alton Meiring and Patrick Mabedi all saw their efforts on goal fail to trouble Baloyi.
Sundowns responded with Oscar Ntwagae shooting wide on 20 minutes, but Baloyi was soon called into action to keep out Cecil Oersen's drive from the edge of the penalty area.
In the 36th minute the veteran 'keeper was at it again, diving to push away Chamanga's shot. A short while later Lefa Tsutsulupa tried his luck from 30 metres out and sent a powerful drive wide of Sundowns' posts, leaving the sides locked a goal apiece at the break.
The Brazilians started the second half brightly, with both Carelse and Bryan Aldave shooting narrowly wide within the first ten minutes after the restart.
However, on the hour mark, Swallows earned a penalty as Shere Lekgothoane brought Oersen down in the box. Oersen himself took the spot kick and calmly stroked the ball into the bottom left of the net to put the Birds 2-1 ahead.
Sundowns struggled to get back into the game after going behind but the visitors did need Etafia to make two good saves within a minute from Lekgothoane and Carelse.
The Nigerian 'keeper made another fine stop in the 87th minute to deny Aldave his second goal in as many games.
In the 90th minute Swallows scored again when Maimane Phiri scored with a direct free-kick leaving Baloyi well beaten.
Swallows move to 10th on 25 points, level with Bloemfontein Celtic and Platinum Stars, while Sundowns remain in a lowly 12th on 21 points.