Celtic finished the match with ten men after Alfred Ndengane was given his marching orders midway through the first half and the visitors were able to capitalize a few minutes later.
Cuthbert Malajila got the first effort on target in the 15th minute when he collected a pass from Teko Modise just outside the box before firing a low shot at Patrick Tignyemb.
Musa Nyatama left the ball for Lerato Lamola at the other end three minutes, who did well to turn his marker, but his weak shot was saved comfortably by Kennedy Mweene.
Tignyemb was the busier of the two goalkeepers and in the 24th minute he did well to stop an effort from Malajila from sneaking into the roof of the net.
Four minutes later, good interplay from Modise and Elias Pelembe set Malajila free on the left side off the box but this time his shot was pushed wide of goal by Tignyemb.
The visitors thought they deserved a penalty in the 43rd minute when Ramahlwe Mphahlele went to ground on the left side of the box under a challenge from Keegan Ritchie but the referee waved play on.
Sundowns had an early chance in the 55th minute when Celtic failed to clear their lines from a throw-in, but Modise drilled his shot wide after he had been set up by Malajila.
Celtic had to play with a man less from the 66th minute as Ndengane was sent off after receiving a second booking.
A minute later Mario Booysen made a good run to meet a cross from Modise at the near post, only to nudge the ball over the crossbar.
However, the opening goal did arrive in the 73rd minute when Malajila took advantage of a defensive error to set the ball up for Pelembe to fire home from close range.
The game opened up from then but it was Tignyemb who had to make key saves to keep his side in the running, diving low to his right to keep out a deflected strike from Modise in the 78th minute.
The Celtic goalkeeper continued to stand his ground between the post, parrying a long range effort from Ejike Uzoenyi in the 88th minute, before denying the same player at the near post in added time.