Gary Mackay-Steven scored twice, with Stuart Armstrong, Chris Erskine, Jaroslaw Fojut and Charlie Telfer also on target for United, while Greg Stewart and Luka Tankulic netted for Dundee.
The Dark Blues suffered a blow during the warm-up to the game when goalkeeper Kyle Letheren injured his thigh and had to be replaced by Arvid Schenk.
And the Dundee substitute was beaten after 36 seconds when Conor Townsend got on the end of Nadir Ciftci’s driven cross and his cut-back was hit first time by Erskine with the ball deflecting off Armstrong and wrong-footing the ‘keeper.
Dundee deservedly levelled the match on 24 minutes when Stewart curled in a wonderful free-kick from 22 yards into the far corner with Rado Cierzniak beaten at full stretch, after Callum Butcher had bundled Gary Harkins over in an aerial challenge.
United restored their advantage three minutes later after Mackay-Steven created some space at the side of the Dundee penalty area by stepping inside Paul McGowan and sending a dipping shot under the bar into the far side of the net.
And the home side struck again on 31 minutes when Townsend’s angled delivery drifted through to Erskine, who stepped into space and sent a low shot which took a nick off a defender on the way past Schenk.
Mackay-Steven scored his second goal three minutes before the break when he beat the offside trap to latch on to an Armstrong pass before beating the ‘keeper with a low shot into his right corner.
And Mackay-Steven turned provider in the 64th minute when he delivered a corner into the six-yard box and Fojut beat James McParke in the air to send the ball inside the near post.
The Dundee defence was posted missing seven minutes from time as Mackay-Steven strode forward onto a through pass and his centre deflected off Iain Davidson into the path of Telfer and the United substitute swept a low shot home from the edge of the box.
While the score flattered United somewhat, Tankulic’s goal in the dying seconds, when he lifted the ball over Cierzniak, will provide little comfort for Dundee.
They had responded well to the early concession and took the game to United, and Harkins produced a neat piece of skill in the middle of the pitch before carrying the ball to the edge of the box and testing Cierzniak with a low drive.
John Rankin came close to getting among the goals for United 10 minutes before the interval when he drove forward from just inside his own half and bent a shot inches wide of Schenk’s left post from 20 yards.
Five minutes into the second half Cierzniak made a brilliant one-handed save to keep out a dipping McPake shot from 25 yards and when the rebound broke to David Clarkson the United ‘keeper blocked his shot with his feet.
And Dundee again came close just after the hour mark when Stewart’s free-kick took a deflection off United defender Keith Watson in an aerial challenge, and Cierzniak tipped the ball off the underside of the bar.
The win for United moved them above Aberdeen into second place in the Scottish Premiership, ahead of the afternoon fixtures.