Kyle Lafferty's fine strike from the edge of the box 11 minutes after half-time sent Hearts to the top of the table with maximum points from their opening two games.
Defeat for Brendan Rodgers's men was also an early season boost for Steven Gerrard's Rangers, who host St. Mirren on Sunday.
Rodgers made five changes with the visit to Edinburgh coming just three days before Celtic travel to face AEK Athens in the Champions League third qualifying round with the tie finely balanced at 1-1.
The visitors struggled throughout a bad-tempered encounter, but did have the best chance of the first period when Leigh Griffiths's shot was cleared off the line by Michael Smith.
Lafferty has been linked with a return to former club Rangers, and showed why Gerrard might be interested with the only goal.
The Northern Ireland international scored three times against Celtic last season and repeated the trick when he met Uche Ikpeazu's cross at the end of a powerful run to drill the ball low past the helpless Craig Gordon.
Rodgers has clashed with the Celtic board in recent days over his lack of backing in the transfer market this summer and defeat will do little to ease the unrest at Parkhead ahead of the trip to Greece.
First-team regulars James Forrest, Tom Rogic and club record signing Odsonne Edouard were introduced in the final quarter in a bid to salvage a point.
Australian international Rogic came closest when he fired inches wide from the edge of the area 15 minutes from time, but Celtic quickly ran out of ideas as Hearts held out for a deserved win.