Derek Lyle's low curling shot was turned past the post by Barry-John Corr, and when Gavin Swankie's corner was floated in, Bobby Mann was on hand to head home from six yards.
Mann then had a header cleared off the line after six minutes before Lyle and Kevin McDonald both saw their headers fly just wide.
Queens levelled against the run of play when John O'Neill scored from the spot after 20 minutes.
It appeared to be a soft award from the ref, but he pointed to the spot when Scott Robertson was adjudged to have fouled his namesake as they chased a through ball.
Andy Thomson had a good chance for the visitors 10 minutes before the break, but his header was poor and Ludovic Roy saved easily.
Dundee took the lead early in the second half when Swankie was fouled by Jim Thomson on the by-line and the ref pointed to the spot once more. Lyle stepped up and hammered his kick low into the corner of the net.
Andy McLaren went close five minutes later, but Corr dived high to his left to tip his header over the crossbar.
Dermott McCaffrey saw his 30-yard shot sail just over the crossbar just after the hour mark. Ten minutes later Swankie curled a free-kick inches past the post.
Substitute Sean O'Connor had a half chance for the visitors late on, but from 12 yards he fired high over the bar.