After missing a first-half spot kick, The Birds struck gold just before the break when Siphelele Ndlovu headed in a well-deserved winner from Lebohang Mokoena's free-kick.
Both goalkeepers, Ajax's Nick Hengelman and Virgil Fries of Swallows, pulled off some neat saves, although the former was far busier during this exciting round 27 fixture.
The result brought Brandon Truter's outfit within three points of the Urban Warriors, who have a slightly better goal difference.
A tired-looking Natal Rich Boyz did well to pull a goal back after AmaTuks' veteran forward Collins Mbesuma opened the scoring in a tight match that almost secures the KZN side's safety for another season.
Abantu Bemthetho, for whom Decide Chauke netted his tenth goal, moved above Mbombela United, who face Uthongathi FC on Wednesday in a fixture that could have repercussions at both ends of the log.