Conversly the win sees the Birds, winners of four of their last five League matches, move into the top eight.
Katlego Mashego scored late on after an excellent performances from both goalkeepers had threatened to condemn the game to a goalless draw.
Bennett Chenene almost put Swallows ahead inside five minutes, letting fly with a long range effort that forced Azwindini Maphaha into a great save.
After that Rodney Ramagalela and Joshua Obaje shot wide with speculative attempts, which were no trouble for Greg Etafia.
Maphaha, in contrast, had to come up with another good save to deny Sibusiso Khumalo.
With the teams trading attacks, Ramagalela fired an effort on target from just inside the penalty area in the 22nd minute, and Etafia made his first notable save to keep the scoresheet blank.
Despite the players being more than willing to shoot on sight, the first half ended goalless.
Early in the second Leopards’ Jean Mugananga continued the trend of long-range attempts, but he was comfortably denied by Etafia, and just past the hour mark Swallows winger Dikgang Mabalane had a shot saved by Maphaha as the keepers continued to dominate the battle.
Both sides brought strikers on off the bench to bolster their attacks – Brian Amidu for Lidoda Duvha and Mashego for the Birds.
Amidu forced Etafia into a great save with his feet in the 70th minute.
With just over 10 minutes to play, the visitors created one of the few close-range chances of the match when defender Larry Cohen got on the end of a corner kick from Lefa Tsutsulupa, but he put his header wide.
The Birds finally broke the deadlock five minutes from time, when Chenene picked out Mashego with a cross from the right, and the striker placed a shot into the bottom corner of the net that even the excellent Maphaha could not keep out.
The late goal proved to be enough for Swallows, who leapfrog Bloemfontein Celtic into the top eight. Leopards remain just one point and one place above the relegation zone.