Pitso Mosimane's men entered the Lucas Moripe Stadium encounter a seemingly insurmountable 13 points adrift of leaders Kaizer Chiefs, but two games in hand meant they at least had the chance to keep the Soweto giants honest.
However, the plan went out the window almost immediately as Lerato Lamola started the onslaught in the very first minute, slotting home a neat finish to leave the hosts stunned.
Celtic wouldn't sit on the lead either and doubled their advantage just seven minutes later, Frank Mhango with his first of two on the night, firing home after Lamola had turned from scorer to provider.
The Tshwane giants were arguably fortunate to go into the break only two goals behind, but that all changed three minutes after the restart as Celtic got their third, Keegan Ritchie setting up Thapelo Morena, who duly put the ball into the back of the net.
It didn't take long for Clinton Larsen's men to added yet another, Musa Nyatama this time getting in on the action after 53 minutes, and it was soon five as Mhango grabbed his brace, squeezing the ball past substitute goalkeeper Kennedy Mweene.
For Sundowns, the result meant they now need nothing short of a footballing miracle to overturn the 13-point deficit to Chiefs, while Celtic moved three points clear in fifth, albeit having now played a game more than sixth-placed Ajax Cape Town.