The Tannadice side were two goals ahead in the opening 14 minutes following strikes from Francisco Sandaza and Feeney before Darren Mackie gave the Dons hope of a point in the second half.
But United held firm to retain third spot in the SPL and stretch their unbeaten run to an impressive nine games, with Aberdeen still failing to leave Tayside with a single point since Craig Levein's arrival.
Such an impressive showing against Rangers midweek meant it was no real surprise to see Levein keep the faith with the same 11 players who were unfortunate to leave Ibrox with just a point.
Sandaza's two goals proved to be enough for him to retain his place in the team at the expense of Roy O'Donovan, who was back in the squad after serving a suspension.
Aberdeen made two changes from the side who defeated Kilmarnock last week. Mark Kerr and Jeffrey De Visscher were drafted into the team in place of Charlie Mulgrew and Andrew Considine, the latter missing out through illness.
United looked dangerous from the outset and Feeney was not too far away with a glancing header across the face of goal after being set up by a cross from the right from Willo Flood.
Sandaza then had the net bulging only for the effort to be disallowed after Feeney was deemed to have fouled goalkeeper Jamie Langfield in the build-up to the strike.
But, with eight minutes on the clock, Sandaza had the ball in the back of the net again and this time there were no complaints from the referee. The move began with a long ball from Sean Dillon before Morgaro Gomis played through Sandaza, who outmuscled Zander Diamond to slot home.
United doubled their advantage six minutes later. This time, Paul Dixon's cross was missed by Sandaza, who failed with a wild swing, before the ball fell to Feeney, who rolled it into the net from close range.
The game should have been all but beyond Aberdeen with 20 minutes gone when Sandaza exchanged a one-two with Flood before seeing his shot blocked by Langfield. The ball fell kindly to the feet of Feeney, who somehow managed to hook over with the gaping net at his mercy.
The Dons had a chance to haul themselves back into the match when De Visscher collected a Sone Aluko ball, only to see his shot swerve wide of target.
A foul on De Visscher just outside the box provided another opportunity to threaten but Lukasz Zaluska did well to touch Lee Miller's well-struck free-kick onto the post before the United defence cleared the danger.
The Dons were still searching for the goal that would give them hope of a point and there was concern for the home side after the restart when Lee Wilkie's attempts at clearing an Aluko cross saw him head dangerously across the face of his own goal.
The defender was involved again at the other end, with a quick free-kick met by Wilkie who nodded on to Darren Dods for a decent strike which was smothered by Langfield.
Mackie then went close for Aberdeen with a powerful header which flashed wide of the upright.
But he was on target with 66 minutes gone when a deep cross from Aluko was nodded back across goal by Miller into the path of Mackie to send his own firm header into the back of the net from close range.
Aberdeen's increasing confidence could have been shattered by another United goal when Scott Robertson slipped a lovely ball through to Sandaza only for the striker to stab agonisingly wide of the post.
But the visitors kept pressing until the end and a cross-cum-shot from Mulgrew was deflected wide by Wilkie as United fought hard to retain their advantage.
Aberdeen thought they had finally claimed the equaliser when Mackie turned a Miller header into the back of the net with just three minutes to go - but the offside flag was already raised.