The visitors were the first to threaten as Tshediso Patjie fired a shot across the Celtic bow after eight minutes, before again getting off an effort four minutes later, this one bringing a fine diving save from Bruce Mtshali.
The goalkeeper was then again needed a few minutes later to keep out Jabulani Dhladhla with his feet, and that save would prove vital as Celtic went on to take the lead with 27 minutes gone, Fiston Abduol Razak slotting home past Malcolm Jacobs in the away goal.
It wasn’t long after the restart that Celtic got their second goal, Razak completing his brace with a smart turn and shot in the area to sneak the ball in past the post.
Thabo Mphahleni looked to work something for the visitors soon after but could only fire his shot wide of the mark, before Siyanda Khumalo saw his shot blocked by Mthokozisi Dube.
The visitors finally found a way through with just three minutes remaining as Jemaine Christian managed to ruin Bruce Mtshali’s clean sheet.
However, there was to be no way back for Mbombela, with Celtic eventually coasting into the quarter-finals.
Celtic: Mtshali, Mahlasela (Shikweni 67'), Rikhotso, Nyatama, Lakay (Hotto 58'), Dube, Letlabika (Nyondo 81'), Sibanda, Abdoul, Phalane, Mvula
Unused subs: Tigmyemb, Mabasa, Molekwane, Sankara
Mbombela: Jacobs, Dhladhla, V. Shongwe, Christian, Nkombelo, Mathibe, (Liphoko 57') Mohlala (Masiya 66'), Maliba, Patjie (Mulenga 60'), S. Khumalo, I. Masuku
Unused subs: Khalishwayo, Mulenga, Sithebe, Ntombayithethi, Mabuza