Not since November 2004 had Dundee got one over their city rivals, but they took advantage of their neighbours' dismal run of form to secure all three points on Wednesday.
United goalkeeper Radoslaw Cierzniak gifted Greg Stewart an opening goal when he allowed his shot to squirm under his legs, but Nadir Ciftci levelled from the penalty spot only two minutes later.
Paul Hartley's side were not to be denied, though, as James McPake put the home side back in front after 23 minutes and Paul Heffernan made sure of a long-awaited derby triumph midway through the second half.
United have now slumped to five consecutive defeats in all competitions, while Dundee are up to sixth.
Celtic moved another step closer in their march towards a fourth consecutive title by beating Patrick Thistle 2-0 at Parkhead.
Kris Commons opening the scoring from the penalty spot after James Craigen had been shown a straight red card for a foul on Stuart Armstrong just before half-time and Stefan Johansen added a second to seal a fifth win in a row in all competitions.
Ronny Deila's men remain seven points clear of Aberdeen with a game in hand after Derek McInnes' men secured a 1-0 victory over third-placed Inverness Caledonian Thistle courtesy of Ash Taylor's strike.
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