The result hoists Phunya Sele Sele into the top eight with 13 points, while the loss for Usuthu means they remain in 14th place with only eight points.
Celtic have been notoriously been slow starters in their last couple of matches and it was no different on Sunday.
The home team were disjointed and struggled to string passes together, allowing AmaZulu to control the early exchanges.
In the 11th minute Mark van Heerden came close to opening the scoring when he curled a 25-metre free-kick around the defensive wall and towards goal, but Patrick Tignyemb produced a great save to deny him.
Two minutes later, though, Tignyemb should have seen red, but luckily for Celtic the referee made the wrong decision.
Sifiso Vilakazi was through on goal and just had the keeper to beat, but as the AmaZulu striker rounded the keeper, Tignyemb appeared to pull him down. The referee waved away the penalty appeals and instead showed Vilakazi a yellow card for diving.
Thereafter, Celtic found their rhythm and created their first chance on goal in the 26th minute.
Ruzaigh Gamildien cut into the box from the right flank, but his low 16-metre shot was well saved by the keeper diving low to his left.
Gamildien was in the thick of the action again three minutes before the half-time whistle when an overhit cross was nodded back into the danger area, but he fired over the bar and the half ended 0-0.
Both sides struggled to create a threat in front of goal in the second half and when they did get in shooting range finishing was poor.
The game looked to be heading for a draw until a deadly finish from Manzini in the 92nd minute handed Celtic all three points.
A deep cross from the left was chested into the path of Manzini by Gamildien and he produced a great finish to steal the win.