Hearts manager John McGlynn was left to rue wasted chances after seeing his side slump to
Matt Lockwood's free-kick was the only goal of the game, with Hearts dropping to ninth in the table.
It was a frustrating day all round for the visitors, who struggled to impose themselves on the game.
"We never got started really," McGlynn told BBC Scotland after the match at Dens Park.
"We found it really difficult to pick it up when we needed to. We could maybe have got a draw out of it in the end but that would've been the most we deserved."
Hearts sealed a penalty shoot-out victory over Dundee United in the Scottish Communities League Cup in midweek and looked jaded throughout against Dundee.
But McGlynn refused to accept that was the cause of the defeat.
"I don't want to make excuses," he said. "That [Wednesday's win] could well have something to do with it but I'm not going to make any excuses.
"We were sloppy at the start and that gave them all the encouragement they needed. When you're in Dundee's situation, that's the kind of start that you want.
"We wanted to put them under pressure and make them nervous but that didn't materialise until too late in the game."
Marius Zaliukas had two good chances to equalise but failed to take the opportunities, a cause for concern for McGlynn.
"In most games we've created many chances, more than we have today," he added.
"We did have opportunities and we might have got something out of it. We certainly need to be more clinical in front of goal."