Dundee lifted themselves from the foot of the table after a match which hinged on two first-half penalties.
The visitors took the lead in the second minute through Ryan Conroy's penalty after Marius Zaliukas fouled Colin Nish.
Hearts passed up a chance to level when John Sutton saw his spot-kick saved by veteran Dundee keeper Rab Douglas.
The home side enjoyed the bulk of pressure in the second period, but could not find a way past Douglas.
The visitors raced into an early lead when striker Nish was pushed in the box by Hearts captain Zaliukas and referee Craig Thomson had no hesitation in pointing to the post.
Conroy stepped up and coolly converted to score the Dens Park side's first SPL goal of the season.
After the early setback, Hearts pushed forward with intent, but this allowed Dundee the opportunity to hit them on the break, and Conroy was unlucky not to connect with a long-range pass with only one Hearts defender between him and goal.
Hearts were showing attacking intent however, and the Tynecastle side were gifted a way back into the game when Lewis Toshney pulled down Ryan McGowan in the box.
However, 40-year-old Dundee keeper Douglas dived low to parry Sutton's spot-kick.
Hearts youngster Callum Paterson then went on a mazy run before letting fly with his weaker left foot with a powerful shot that drifted off target.
The home were in the ascendent as the first half drew to a close, and Douglas again got down low to save from Dale Carrick.
New signing Ryan Stevenson made his return to the Hearts side when he came on as a second-half substitute for Sutton.
After a quiet opening spell of the second half, mercurial winger Arvydas Novikovas carved out a chance for Paterson, but the youngster couldn't get over his shot and it went over.
Hearts were dominating, and Stevenson lashed a shot from distance after Davide Grassi slipped in possession which stung Douglas' gloves.
From the resulting corner Conroy cleared Stevenson's header off the line, with Carrick blazing the follow-up over the bar.
Grassi misjudged a Danny Grainger cross and almost turned the ball into his own net as the home side continued to press, but he was the only threat to Douglas' goal with no Hearts player in the box.
As the clock ticked down, Hearts' attacks became more desperate, but, as a result, they lost composure in the final third and the Dark Blues held on to claim a first league win and lift themselves off the foot of the table.[an error occurred while processing this directive]