Hearts defender John Souttar headed an 89th-minute winner as Ange Postecoglou's difficult start to his Celtic reign continued into the Scottish Premiership season.
Souttar rose to head home Michael Smith's free-kick and earn the hosts a 2-1 victory in front of 5,272 fans at Tynecastle.
Gary Mackay-Steven had given Hearts an early lead as Swedish defender Carl Starfelt endured a torrid opening to his Celtic career, 24 hours after his first training session.
Anthony Ralston netted an impressive equaliser in the 54th minute but the home side responded well and Souttar consigned Postecoglou to a second defeat this week following Celtic's Champions League exit against Midtjylland.
Celtic's Old Firm rivals Rangers made a winning start to their Scottish title defence earlier on Saturday, beating Livingston 3-0 at Ibrox courtesy of goals from Ianis Hagi, Scott Wright and Kemar Roofe.
Both Starfelt and Nir Bitton came into the Celtic central defence and James Forrest made his first start of the season, while Hearts handed a debut to Beni Baningime after his arrival from Everton.
Celtic have been accustomed to opening their season at home but they had to play a supporting role as Hearts raised the Championship flag.
The visiting players had welcomed their opponents on to the park with a guard of honour and they were overly generous with their defending soon afterwards.
Starfelt made a fresh-air clearance inside 60 seconds and the Swede looked suspect again as Hearts took the lead in the eighth minute.
The new arrival from Rubin Kazan failed to halt Josh Ginnelly's progress in an attempted tackle and then went to ground in a desperate attempt to stop Liam Boyce capitalising. The ball ultimately broke for Mackay-Steven to stroke home from six yards.
David Turnbull squandered Celtic's best first-half chances, twice failing to hit the target in space on the edge of the box and also hitting a free-kick into the wall.
There were other half-chances for the visitors. Craig Gordon comfortably saved Odsonne Edouard's header and long-range efforts from Callum McGregor and Ismaila Soro, while Liel Abada failed to test the goalkeeper after twice cutting inside.
But Hearts had opportunities to go into the break further ahead.
Liam Boyce headed not far over from a corner and the hosts were getting in behind the left side of Celtic's defence. Michael Smith fired into the stand when Boyce was well-placed in the middle before setting up fellow wing-back Alex Cochrane to volley wide.
Celtic levelled after some positive play from Ralston, whose only appearance last season came against Livingston in January when 14 of his team-mates were in self-isolation.
The defender got into the type of central attacking position Postecoglou demands of his full-backs, played a one-two with Edouard before swerving round an opponent and curling home with his left foot.
It was the 22-year-old's first goal since December 2018 and the first goal Gordon had conceded in 12 club matches.
Hearts had been pinned back for much of the half but the introduction of midfielder Peter Haring turned the tide and Scott Bain had to scramble to stop Souttar's header.
Postecoglou responded by putting on Ryan Christie and debutant Kyogo Furuhashi, who had not even had time to train after his quarantine.
The double switch almost paid immediate dividends with Gordon saving well from his Scotland team-mate.
But Hearts were soon back on the front foot and Souttar headed past Bain with the goalkeeper stranded.
Gordon denied his former side with two excellent stoppage-time stops from Christie and Starfelt.