Conroy's third-minute penalty was all that was required for Dundee to secure all three points and lift them off the bottom of the table.
Hearts dominated the attacking possession but couldn't beat an inspired Rab Douglas in the Dundee goal. The veteran goalkeeper made a string of impressive stops including John Sutton's first half spot-kick.
The hosts made one enforced change to the team that drew 1-1 with Liverpool in the Europa League play-off on Thursday. Anfield scorer David Templeton completed a switch to Rangers on transfer deadline day and was replaced in the starting line-up by Dale Carrick, the youngster making the first start of his professional career. New signing Ryan Stevenson had to be content with a place on the bench.
Visitors Dundee made two changes to the side that exited the League Cup at the hands of Queen's Park, as
striker Colin Nish was recalled to the starting line-up in place of Steven Milne while Graham Webster dropped out to make way for the fit-again Lewis Toshney.
Nish made an immediate impact by helping to give Dundee the lead inside three minutes. The former Hibs striker turned into the penalty area and was needlessly tripped by Hearts captain Marius Zaliukas. Referee Craig Thomson showed no hesitation in pointing to the penalty spot, and up stepped Conroy to smash the ball low past Jamie McDonald.
Hearts pressed their opponents looking for a quick route back into the match, but they had to wait 20 minutes for their first sight of goal. Even then it was a speculative effort as Callum Paterson's 30-yard attempt was cautiously pushed over the bar by Douglas.
The home side were then rewarded a route back into the match as Ryan McGowan wriggled his way through the penalty area before being tugged down by Toshney. Again, Craig Thomson immediately awarded a penalty and although Sutton's spot-kick was hit with power and accuracy, it was brilliantly parried away by Douglas diving low to his right hand corner.
Paterson showed pace and power to leave two Dundee defenders in his wake, but scuffed his shot from the edge of the box on his weaker left foot.
With the first half drawing to a close, fellow teenager Carrick then went close to restoring parity. The attacker's swivelled shot from 18 yards had to be tipped round the post by a stretching Douglas.
For the second half, Hearts boss John McGlynn introduced Stevenson for his second Hearts debut in the hope of injecting life into his side's struggling attack. Withdrawing was Sutton in a like-for-like swap in terms of position and experience, but it was the youngsters who again went closest to levelling the match. Novikovas' cutback found Paterson in the penalty area but he fired wide from 10 yards.
It wasn't long, however, before Stevenson began to influence proceedings. On the hour he produced a save from Douglas and then from the resulting corner he beat the keeper with a glancing header, only to see Conroy clear the ball off the line.
For the final 25 minutes Hearts were relentless in their pursuit of a goal but couldn't find their way past a Dundee side that defended strongly in the face of a continuous wave of attacks.
For all Hearts pressure there were few chances and the game finished with nothing to show for their efforts.