Dennis Wyness pounced on an error in the Dundee defence with five minutes gone, but the striker skied his effort over the crossbar from 8 yards.
A shot on the turn from 12 yards by Tam McManus went just wide of the target after good play by Neil Barrett on 19 minutes.
Fabian Caballero then hit a weak shot straight at visiting keeper Craig Gordon on the half-hour mark.
Caballero's next effort was a right foot shot, which was more threatening but deflected over the crossbar for a corner.
The Dundee striker wasn't to be denied, however, and he put his side ahead after 54 minutes when he headed a poor clearance by Jamie McAllister past Gordon for his first goal since 2003.
The home team's lead was short-lived and Hearts substitute Joe Hamill drew his team level six minutes later with a low ten-yard drive, just moments after coming off the bench.
John Sutton had a great chance to put Dundee back in front 20 minutes from time, but from Lee Wilkie's headed knockdown, the big striker volleyed over the crossbar from four yards.
Both teams had decent efforts on goal as the game went into the last ten minutes.
Firstly, Mark Fotheringham's 25-yard free-kick was saved by Gordon before Hamill thundered a right-foot shot inches over from outside the box.
Sutton got his head to Stephen McNally's cross in a crowded penalty area, but the big striker could only direct the ball over the crossbar as the three minutes of time added on ticked away.