Aces prevailed through strikes from Abia Nale, Kwabena Adusei and Eddie Ngalo, while Larry Cohen and substitute Vuyisile Wana were on target for the struggling Birds.
The win leaves AmaZayoni sixth on the Absa Premiership table with 12 points from eight games - nine points behind leaders Kaizer Chiefs, who visit Ajax Cape Town tomorrow in a top-of-the-table clash in the Mother City.
The Dube Birds, meanwhile, are 11th in the standings with nine points, having lost five games in a row in all competitions.
Aces captain Nale gave his side the lead in the 25th minute after a one-two with Mabhuti Khenyeza and finish from the edge of the box.
Swallows, though, hit back in the 40th minute when Cohen scored his first goal for the Birds when he got onto the end of a Dikgang Mabalane corner, slotting the ball home after the goalkeeper parried Anver Esterhuizen's header into his path.
Aces, however, took the ascendancy into the break with virtually the last kick of the opening half after Adusei headed home an Onyekachi Okonkwo corner-kick, 2-1.
The away side doubled their lead in the 60th minute after another well-worked set-piece saw Ngalo head home another Okonkwo delivery after Absa Man of the Match Adusei knocked the ball back across goal, 3-1.
Swallows, though, were not going to down without a fight and pulled a goal back two minutes later thanks to long-range wonder strike from Vuyisile Wana.
Aces set up their stalls in defence after that and defended like soldiers to hold out for the win.