Hearts substitute Mark de Vries earned his side a point with a well-taken strike 16 minutes from time.
Aiden McGeady celebrates his debut goal
Celtic had taken the lead in the first half courtesy of a debut goal from 18-year-old Aiden McGeady.
The youngster lashed the ball past Craig Gordon from 10 yards after Stilian Petrov's free-kick was missed by the head of Stan Varga.
But Hearts battled gamely and De Vries equalised from a similar position to the one McGeady had scored from.
Both sides started the game in spirited fashion, with some crunching tackles flying in, though chances were at a premium in the early stages.
McGeady's dream goal came after 17 minutes, but it was entirely avoidable from Hearts' point of view.
McGeady was fouled needlessly by Alan Maybury on the right flank and when Petrov's free-kick was missed by everyone in the middle, the youngster - unmarked at the back post - drove a low shot in off Gordon's right-hand upright.
Andy Kirk passed up a tremendous opportunity to level the scores after 32 minutes.
Sent clear by Patrick Kisnorbo, Kirk hit his first effort straight at David Marshall and when the ball fell kindly for him, he blasted the rebound over the bar with the goal gaping.
Ten minutes later Kirk had another glaring miss when Andy Webster nodded Paul Hartley's corner down to him only a few yards out, but the Northern Ireland striker failed to connect properly with the ball and Marshall gratefully gathered it on the line.
The second half started in the same manner as the first, plenty of blood and thunder in the midfield, with little in the way of goalscoring chances.
De Vries celebrates his equaliser
Celtic came close to a second goal after 68 minutes when Petrov sent a viciously curling free-kick into the box and Henrik Larsson was only inches away from connecting at the back post.
Three minutes later, the same pair combined at a corner kick and the Swede glanced a header narrowly wide.
But Hearts, with almost their first effort on goal in the second period, pulled level after 74 minutes.
Johan Mjallby took a fresh-air swipe at a Scott Severin cross and De Vries thumped the ball past the helpless Marshall and into the net.
Hearts looked the more likely to add to their tally in the latter stages and Hartley almost caught out Marshall at his near post with a clever free-kick that the keeper did well to turn away.
Severin then chipped the ball just over the bar, with Marshall caught off his line, before the young keeper saved superbly from De Vries' powerful long-range effort.
Hearts: Gordon, Pressley, Webster, Severin, Neilson, Stamp,
Hartley, Kisnorbo, Maybury, McKenna, Kirk.
De Vries, MacFarlane, Weir, Hamill.
Celtic: Marshall, Agathe, Varga, Balde, Mjallby, Miller,
Lennon, Petrov, Smith, McGeady, Larsson.
Subs: McGovern, Sylla,
Lambert, Beattie, McManus.
Referee: K Clark