Amakhosi immediately proved why they deserved to be in first place when they found the back of the net in the seventh minute.
Swallows' defence could not cope with the speed of Siphiwe Tshabalala and Majoro, who combined beautifully before the latter sliced in a cross from the edge of the box to meet Reneilwe Letsholonyane, who tapped it in with ease.
15 minutes in the Dube Birds had a rare break in Chiefs' final third and Katlego Mashego curled a free-kick in at the far post but it was dealt with by Itumeleng Khune.
The game started to pick up some pace and flare and it was Swallows who pulled themselves right back into it, when Lerato Chabangu calmly drove a low effort inside at the far post after some great build-up play.
The Soweto Derby atmosphere was electric, and it was heightened five minutes later, when 'Yeye' completed his brace. Bernard Parker was the man supplying the pass, and it was a deadly assist that made it all too easy for Letsholonyane to hit home from inside the box.
Swallows needed to make a comeback fast at the start of the second half, and they nearly did, when Bennett Chenene went on a great run on the flank, cutting into the box and rounding the defenders, only to see his attempt denied by Khune.
Chenene caused further trouble for Chiefs when he handed Swallows a deserved equaliser on the hour mark, hitting home a snap-shot from an excellent cross, supplied by Giorgi Nergadze.
Stuart Baxter responded by making a double substitution, bringing on Kingston Nkhatha and Siyabonga Nkosi.
Soon after that Greg Etafia pulled off an excellent save with his right foot to deny Tshabalala the winner.
Etafia and Nkosi collided badly in the final 10 minutes of the game but both managed to continue playing.
Seven minutes of stoppage time were added at the end of the game, and it was Chiefs who capitalised when Majoro headed home the winner in the final minute.