A high-speed chase through the Cape Flats ended in a gunfight after a car rammed pursuing traffic co.
With the pressure mounting, Sundowns looked anxious through the first half but Mame Niang unshackled them early in the second as weight was lifted off their shoulders after they had failed to win in their three opening encounters of the season.
Niang proved he is not only a target man with some clever ball skills to beat Greg Etafia for the opening goal in the 53rd minute.
It came after a silly mistake at the back from Swallows, which they replicated to gift Sundowns a second.
Substitute Kudakwashe Mahachi scored with a side-footer for a second in the 83rd minute as Swallows ended the game down to 10 men.
A second caution for Lantshene Phalane let down his side just when they needed to push forward.
The Birds had a disciplined approach through the first half as they sat back and created a defensive wall that Sundowns found hard to penetrate.
For all their possession, the champions found themselves increasingly hurried in their decision-making as the half unfolded and they made limited headway.
When Tebogo Langerman got around the defence in the 26th minute, his attempt to set up Niang was blocked.
Etafia fluffed a corner to almost present Sundowns with the first goal on the half-hour mark but was rescued by Tsietsi Mahoa's clearance off the line.
Clever play from Teko Modise and Themba Zwane set Modise up for a shot right in front of goal in the 40th minute, but he shot wildly.
After the break, Modise pushed closer to Niang and Sundowns looked calmer, and once they scored were able to put a smile back on their faces.