Clive Moyo-Modise and Noah Chivuta, who also missed a penalty, scored for the Students, while Celtic saw defender Keamogetse Wolff sent off early in the first half.
The game got off to a crazy start with Wits’ skipper Moeneeb Josephs receiving a yellow card before the match kicked off, for what seemed like the goalkeeper’s delaying of the kick off.
Wolff then received his marching orders for a two-footed challenge on Wits’ striker David Radebe in the seventh minute.
The hosts were awarded a penalty in the 21st minute after Celtic defender Gary Goldstone was adjudged to have fouled Moyo-Modise inside the area, but Chivuta sent his spot-kick crashing against the crossbar.
Moyo-Modise made it 1-0 in the 41st minute when he tapped home from close range after Celtic stopper Postnett Omony spilled a shot from Marawaan Bantam.
The score remained 1-0 at half-time.
Josephs produced a brilliant save in the 59th minute to deny Kleinbooi Taaibos from hitting the back of the net.
The Students’ skipper continued his heroics in the 70th minute when he denied Thabo Nthethe with another world-class save.
Chivuta put the result beyond doubt in stoppage time when he scored from close range after Omony again spilled a Bantam cross.