The former champions needed maximum points from a fixture delayed for 72 hours by a waterlogged pitch, to knock arch-rivals Orlando Pirates off the summit on goal difference.
But Chiefs were grateful to grab a point via a Knowledge Musona goal two minutes into additional time, after falling 2-0 behind less than a third of the way through the opening half.
Moses Spandeel and on-loan Lennox Bacela scored for Celtic within 14 minutes of the kick-off before a capacity crowd crammed into Seisa Ramabodu Stadium, one of the training grounds for the 2010 World Cup.
Steven Ncanana took time off from his defensive midfield duties to pull one goal back six minutes before half-time, with a low shot into the corner, before leading league scorer Musona raised his goal haul to 10 this season.
The result saw Chiefs climb one place to third, above Ajax Cape Town on goal difference, two points adrift of 1995 African champions Pirates and one behind title favourites Mamelodi Sundowns.
Celtic stayed sixth, below Santos on 26 points, and with 14 rounds to go in the richest national football championship on the continent, they will believe the 1.3-million-dollar first prize is not out of reach.
"My players gave it all and I cannot fault them for effort or application. We wasted scoring chances when two goals ahead and this encouraged Chiefs to fight back," admitted Celtic coach Clinton Larsen.
Chiefs Serb coach Vladimir Vermezovic hailed his team: "We had nothing to lose in the second half and sometimes put five players up front. I am satisfied with a point, and the last five rounds will decide who wins the title."
A mid-season revival by defending champions SuperSport United came to a shock halt when they lost 2-1 after taking the lead at bottom-of-the-table Mpumalanga Black Aces in north-east mining town Witbank.
Aces defender Joseph Henyekane headed a corner into his net midway through the first half, David Mathebula levelled on the stroke of half-time, and new signing Manqoba Ngwenya headed the winner nine minutes from the final whistle.