Tumelo Njoti, who starred in the last round against AmaZulu, returned for the hosts as did Maloisane Ramasimong. For Maritzburg, Bongani Mpandle and Jeremy Brockie were drafted back into the line-up.
Celtic invaded opposition territory straight from kick-off and missed the chance to go ahead after Lantsane Phalane's long-range drive rattled the post despite Mpandle getting a slight finger-tip.
It was United though that conjured up the opener albeit against the run of play as Thabiso Kutumela fired home from the penalty spot on 16 minutes after Justice Chabalala was flagged for handling in the area.
Celtic were level within seven minutes as miscommunication between Mpandle and Miguel Timm allowed Njoti to head into an empty net following a loose ball in the area.
Ramasimong almost handed his side the lead with an acrobatic overhead kick seven minutes before the break, but Mpandle managed to go down and avert the danger.
Celtic missed a golden chance to take the lead a few minutes from the interval as Mzwanele Mahashe headed over the top from the far post following a corner kick.
The Team of Choice were behind for the first time in the game after poor handling from Mpandle found a lurking Ramasimong, who tucked home the rebound on 64 minutes.
Bongokuhle Hlongwane should've found the leveller for the visitors with a close-range shot off a corner two minutes later, but Chabalala pulled off a goal-line clearance.
The Team of Choice grabbed the equaliser in the final two minutes after Kwanda Mngonyama volleyed home a rebound from Jackson Mabokgwane's one-handed palm off a corner kick.
With extra-time approaching, Siwelele netted the winner in time added on after Harris Tchilimbou tapped in off the post to guide his into the last eight.
Celtic: Mabokgwane, Rikhotso, Ngobeni, Maloisane, Mahashe, Njoti, Chabalala, Baloyi, Phalane, Potsane, Mabena
Subs: Chaine, Maema, Tani, De Jongh, Luthuli, Sera, Tchilimbou.
Maritzburg: Mpandle, Sanoka, Mngonyama, Xulu, Kunene, Kutumela, Timm, Claasen, Ntshangase, Tlolane, Brockie.
Subs: Ofori, Shandu, Buchanan, Hlongwane, Jappie. Morgan, Nyoni.