The Birds are now on 50 points, two behind log leaders Orlando Pirates with two matches to play.
Swallows, the formerly dormant Soweto giants, have lost just four matches in the league, the fewest of any team.
All seven goals – including three braces – came in a frenetic last 33 minutes after scores were deadlocked at 0-0 at the halftime interval.
Georgi Nergadze scored the opening goal with a header in the 57th minute, after which the floodgates opened.
Nomvete, Mpho Maleka and Kingston Nkatha all came away with braces.
'Bhele' now has 18 goals this season, and should get to the 20-goal mark if he keeps his form up.
Maleka came off the bench to add two of his own, before Nkatha found two consolation goals in the last five minutes of play.
Gordon Igesund's men have lost just four matches this season – the lowest number of losses in the League.
Leopards did well to keep the visitors at bay in the first half, and Lerato Chabangu and Nomvete had half-chances.
Lidoda Duvha's best chance of the half came from Tiyani Mabunda, who had a shot tipped over the crossbar by Greg Etafia just under half an hour in.
But the hosts ran out of steam in the second half, and as tensions ran high coach Vladislav Heric and technical advisor Sunday Chidzambwa looked to be in tactical disagreement.
Leopards have three matches in hand, but are in trouble in 14th place with 24 points, just one above Santos.