Goals from Ryan Dow, Gary Mackay-Steven and Chris Erskine secured the points for Jackie McNamara’s side, while Aberdeen paid the price for a pair of first-half defensive mistakes.
The Dons were first to threaten but Andrew Considine volleyed over the crossbar after a Niall McGinn cross had been flicked on by Ash Taylor.
Callum Morris headed wide at the other end before McGinn tried his luck, only for Radoslaw Cierzniak to make a comfortable save.
And Aberdeen then found themselves a goal behind just before the half-hour when Taylor’s back-pass sold Jamie Langfield short and Dow (29) pounced to shoot past the home goalkeeper at his near post.
The hosts were within inches of equalising just before half-time, but the dangerous McGinn’s volley hit the bottom of the post after Cierzniak had failed to deal with a Barry Robson cross.
And the visitors made the most of their reprieve, doubling their advantage on the stroke of half-time when Robson was caught in possession and Nadir Ciftci played in Mackay-Steven, who beat Langfield with a low shot.
Aberdeen almost got themselves right back into the match just after half-time but Dundee United’s Conor Townsend blocked Peter Pawlett’s effort after David Goodwillie had done well for the hosts.
Ciftci then had a good chance for the visitors but blazed his shot high over the crossbar.
The Dons fans clamoured for penalties on a couple of occasions in the second-half, notably when Adam Rooney was challenged by Cierzniak and when a cross appeared to hit Morris on the hand, but the referee was unmoved.
Goodwillie then had a late chance but headed over after Cierzniak failed to punch clear a Nicky Low cross.
And Dundee United scored a third with the last kick of the match, Erskine shooting into the bottom corner after good work from his fellow substitute Mario Bilate.