The Absa Premiership leaders were undone by goals from Lyle Lakay, Lantshene Phalane and Fiston Razak, who is on loan at Celtic from Sundowns.
Defender Wayne Arendse pulled one back for the Brazilians, who had Hlompho Kekana sent off in the final minute for a second bookable offence at Lucas Moripe Stadium.
With second-placed Bidvest Wits beating Orlando Pirates 1-0 on the night, Pitso Mosimane's side saw their lead at the summit get shaved to four points with seven matches left.
The shock result also eased the pressure on Phunya Sele Sele coach Serame Letsoaka as it snapped a five-game losing run in all competitions for the visitors.
Letsoaka's side, who remain 12th in the standings, showed their intentions five minutes in when Dumisani Zuma broke free and took a shot, but goalkeeper Denis Onyango made a save.
Lakay scored soon after when Sundowns lost the ball in the middle of the park. Skipper Thabo Nthethe couldn't clear the ball and Lakay beat the oncoming Onyango easily, making it 1-0.
As Mosimane's men tried to respond swiftly, Khama Billiat's cross failed to reach Leonardo Castro in the box and Teko Modise also tried a cross in the ninth minute which looked like it was going in, but Patrick Tignyemb fisted it over for a corner.
As the rain continued to bucket down, Siwelele kept their hosts' defence busy although Thapelo Morena couldn't connect with the ball correctly in the 16th minute.
In the 26th minute, the log leaders received a penalty when Billiat was fouled in the box by Wandisile Letlabika, but Modise missed a chance to level matters as Tignyemb guessed right.
With 10 minutes remaining to the interval, Castro's header went wide as he targeted the near post, and in turn Sundowns gave Zuma more space as he made his run towards the box, but he was let down by his finish.
Nevertheless, Siwelele continued to probe for a second goal and they were rewarded with four minutes to go to the break when Razak got behind the defence and beat Onyango after a sweet pass by Morena.
Phalane made it 3-0 in the 51st minute as he pounced on a loose ball before unleashing a shot past Onyango at his right post.
Towards the 60th minute, Kekana tried a long-range shot but it went wide of Tignyemb's posts after a brilliant lay off by Keagan Dolly from the right flank.
The Brazilians kept chasing a goal and Celtic finally cracked when Arendse converted Kekana's cross with a fine header from the left wing in the 67th minute.
Sundowns dominated proceedings in the last 15 minutes but Siwelele stood firm at the back. In the dying minutes of the tie, the log leaders suffered a blow when Kekana was shown his second yellow card after a foul on Phalane in the middle of the park.