Celtic instituted four changes for the encounter as Mzwanele Mahashe, Ronald Pfumbidzai, Kabelo Dlamini and Tebogo Potsane came into the side.
Potsane repaid the faith shown in him, with a well-taken finish from slack defending inside three minutes to give Celtic an early lead.
Terrence Dzvukamanja managed to get in on the action following a cross by Keagan Ritchie, but the Zimbabwean failed to profit on nine minutes.
Wits looked to get a bail-out after Tshepo Rikhotso conceded a penalty in the 37th minute for fouling Gabadinho Mhango. Ritchie though failed to beat Tignyemb who made a save.
In the second half, Celtic finally succumbed to the intense pressure from the visitors who drew level courtesy of Mhango on 58 minutes before Deon Hotto hit the crossbar two minutes later.
Celtic threatened to restore their cushion after 64 minutes but Wits goalkeeper Darren Keet saved expertly well off Kabelo Dlamnini’s shot.
Nine minutes from full-time, Motupa put the travelling side in front after heading home Thabang Monare’s cross, but Celtic restored parity once again with four minutes to go as Tshegofatso Mabasa finished from close-range.
However, Dzvukamanja restored the lead for Wits minutes later and added a fourth at the death to complete his brace and seal a 4-2 win over the title chasers.
Celtic: Tignyemb, Mahashe, Pfumbidzai, Rikhotso, Mabasa, Tani, Maema, Mashikinya, Dlamini, Potsane, Baloyi.
Subs: Dambe, Letsoalo, Mabena, Masuku, Lingwati, Sam, Shikweni.
Wits: Keet, Mere, Mkhwanazi, Nonyane, Hlanti, Scott, Alexander, Ritchie, Mhango, Dzvukamanja, Majoro.
Subs: Goss, Weidlich, Hotto, Monare, Motupa, Pelembe, Murray.