The key moment at Palmerston Park came in the 44th minute when Queens goalkeeper Jim Atkinson was dismissed for a professional foul. Dundee's Ryan Conroy dispatched the subsequent penalty kick.
Iain Russell struck the crossbar in the 62nd minute, but Dundee survived that scare and would have increased their winning margin had Christian Nade's strike not rebounded off the hosts' upright in the 75th minute.
Promotion challengers Falkirk slipped to third place as they threw away a first-half lead to lose 2-1 at Ian Murray's in-form Dumbarton. Rory Loy fired the visitors ahead from close range in the 29th minute, the striker notching his 14th goal of a prolific campaign.
However, Dumbarton drew level shortly after half an hour as Colin Nish crossed for former Aberdeen striker Mitch Megginson to finish emphatically amid woeful Falkirk defending.
It went from bad to worse for the visitors when, in the 73rd minute, Jordan Kirkpatrick took his turn to prosper from another defensive lapse and crashed the ball home from close range.
Hamilton Academical climbed to second place after a 4-2 triumph at Scottish Cup quarter-finalists Raith Rovers. Tony Andreu netted the Hamilton opener after only three minutes from 30 yards and he doubled his tally just after the half-hour mark.
Louis Longridge then fired Hamilton into a three-goal advantage before James Keating made it 4-0 from the penalty spot before half-time.
Crestfallen Rovers did reduce their arrears in the closing stages with goals from Joe Cardle and John Baird.
Morton are still firm favourites for relegation even though they came from behind to salvage a 1-1 draw with Cowdenbeath at Cappielow.
John Armstrong headed Jimmy Nicholl's men ahead from a corner kick after only 13 minutes.
Garry O'Connor drew two fine saves from Cowdenbeath goalkeeper Thomas Flynn as Morton attempted to launch the fight back to save the match and their season.
A deserved equaliser did eventually arrive in the 85th minute as a cutback was cleverly swept home by substitute Archie Campbell to extend struggling Morton's unbeaten run to five games.
Alloa Athletic's new manager Barry Smith suffered a disappointing 3-0 defeat at the hands of Livingston at Recreation Park on his league debut for the Wasps.
Martin Scott set up the robust Simon Mensing for Livingston's opening goal in the 20th minute, but that was to be the pinnacle of his day.
A crude challenge on Alloa's Edward Ferns in the 60th minute earned Mensing a deserved red card, but his team-mates spared his blushes as Livingston added to their advantage in his absence.
Scott doubled the visitors' tally with a sensational long-range strike in the 81st minute. Marc McNulty then set up Keaghan Jacobs to score Livingston's third to compound the misery for Smith's side who now sit four points above the relegation play-off spot.