Icelandic international Kari Arnason opened the scoring when he fired the ball into the roof of the net from close range after Rory Fallon's header had been kept out by Dusan Pernis.
A Youl Mawene header six minutes after the break gave the Dons a second before Andrew Considine added a third when he slotted the ball home at the back post.
United substitute Lauri Dalla Valle pulled a goal back, but Aberdeen held on for all three points.
Aberdeen stuck with the same team that thrashed Dunfermline 4-0, with Scott Vernon - the hat-trick hero of the SPL's Friday night football experiment - again partnered New Zealand international Fallon in attack.
The Terrors, looking to bounce back after a 3-1 home defeat to Motherwell, made two changes.
Youngsters Johnny Russell and Scott Allan were both handed starts in place of the injured Keith Watson and Ryan Dow, who dropped to the bench.
Dundee United made an encouraging start with Russell's snap-shot from the edge of the box drawing a fine save from David Gonzalez after only five minutes.
Aberdeen skipper Richard Foster was next to try his luck with a curling free-kick which was well held by Pernis.
The Dons made the breakthrough after 14 minutes from a Fraser Fyvie corner when Fallon's downward header was well kept out by Pernis, only for midfielder Arnason to scoop the ball into the roof of the net from close range.
The home support were appealing for a penalty after 28 minutes when Rob Milsom tumbled in the box under pressure from United's makeshift right-back Willo Flood.
The Dons came close to doubling their advantage a couple of minutes later when Fallon was unable to turn the ball home after a tantalising cross from Fyvie.
United had a penalty appeal of their own turned down before the break when Danny Swanson looked to have been clipped by Ryan Jack, but referee Willie Collum waved away the protests.
Swanson failed to recover from the challenge and was replaced by Dalla Valle after 41 minutes.
The Dons carved out an early opening within minutes of the restart when Arnason laid the ball into the path of the onrushing Vernon, but the Englishman's effort was blocked by Flood.
Mawene increased Aberdeen's lead after 51 minutes when he rose highest to head home Foster's free-kick to the delight of the home support.
And Mawene's central defensive partner Considine squeezed the ball home at the back post on the hour mark for Aberdeen's third after the United rearguard failed to deal with a Milsom free-kick into the box.
United pulled a goal back after 78 minutes when Stuart Armstrong, who had replaced Scott Allan, drove the ball across the face of goal for Valle who had the easiest of finishes.
Despite a late spell of United pressure, Aberdeen held on to record back-to-back wins for the first time in the Clydesdale Bank Premier League this season.
