A Mario Bilate penalty put the visitors in front and Ryan Dow’s calm finish gave them control of a game which they never looked like losing.
Headers from Callum Morris and Keith Watson, both from Paul Paton free-kicks, put the seal on a strong performance and win, with Greg Stewart scoring a late consolation for the hosts.
Only in the final 15 minutes did United dominate proceedings but they punished Dundee for their mistakes to extend their unbeaten run in the derby to a seventh game.
The contest sprang into life on 24 minutes, when Gary Mackay-Steven was brought down in the box by Thomas Konrad to give the visitors the perfect chance to open the scoring.
Bilate made no mistake from the spot, sending home goalkeeper Kyle Letheren the wrong way with an emphatic finish from 12 yards.
The response from Dundee was a positive one, with Phil Roberts opting for placement over power when he side-footed Gary Harkins’ deep left-wing cross narrowly wide of the target.
Jim McAlister headed a looping Greg Stewart free-kick onto the roof of the net from 12 yards and it was to prove a costly miss as 10 minutes after the restart, Dow had doubled United’s advantage.
McAlister was caught in possession 10 yards from the edge of his own box, losing the ball to Dow, who advanced with it and kept his head before steering a cool finish past the right hand of Letheren.
The result was put beyond all doubt on 58 minutes when Paul Paton’s excellent inswinging free-kick was headed home by Callum Morris from no more than five yards.
Morris’ contact was scruffy but the quality of the delivery meant he did not have to meet the ball perfectly to put it past Letheren.
Paton again provided an exquisite delivery from the dead ball to provide the cross for Watson to power home a header at the back post as Jackie McNamara’s men began to take full control of proceedings.
Dundee’s agony was compounded in the final minute of the game when Konrad was shown a second yellow card for a late challenge on substitute Scott Smith.
Stewart ensured Dundee were represented on the scoresheet with a left-footed shot from distance but it was almost the final kick of the game and no more than a consolation.