Mame Niang opened the scoring on 10 minutes, but Katlego Mashego equalised 11 minutes later. Niang added another ahead of half-time, before Bennett Chenene made it two all. Mpho Maruping scored eight minutes from time to hand Tuks the win.
Swallows' slack marking in the first half allowed Niang to score twice, both times winning aerial battles to do so.
The visitors certainly started the better side and were a little unlucky not to take the lead in only the second minute of the match as Niang fired centimetres wide from 20 metres out.
However, the striker did not have to wait long to find his first goal of the night when he headed home a Lionel Mutizwa cross in the 10th minute.
Mashego levelled matters, turning home a Lerato Chabangu cross, but Swallows' joy was short lived as the Premiership rookies outclassed them in the air for a second time six minutes before the break.
Again Niang was on hand to rise above his marker and sent a powerful header into the net to make it 2-1.
For all their possession in the second half Swallows struggled to create a decent scoring opportunity in front of goal.
Cue Mpho Maleka, who let rip from just outside the box – Washington Arubi was up to the challenge, making a fantastic save, but he parried the ball straight into the path of Chenene who tapped home from close range.
Finally, some sloppy defending from Swallows enabled Maruping to find space inside the 18-yard area and he made no mistake, slotting the ball home to give his side the lead for the third time.