A Gavin Gunning header after 14 minutes put Peter Houston's men ahead before Russell netted twice in three minutes just before the break to give the Terrors an unassailable lead.
Both sides had to play the final 20 minutes with 10 players after Russell and Dundee captain Stephen O'Donnell both saw red for violent conduct.
The win moved United to the top of the SPL as they maintained their 100% record, while Dundee were left still looking for their first victory and goal of the new campaign.
United made two changes as they took to the field for the first time since a 5-0 thrashing against Dinamo Moscow in a Europa League qualifier with Michael Gardyne starting in place of the injured Gary Mackay-Steven, while Sean Dillon was preferred to Brian McLean in defence.
Dundee brought in Nicky Riley, who had returned from a hamstring injury, at the expense of Mark Stewart in their only change.
New signing Martin Boyle was named on the bench after joining from Montrose on Thursday.
The game started at a frantic pace with a Gardyne shot bravely blocked by Dundee defender Matt Lockwood before Riley was twice denied at close range by United stopper Radoslaw Cierzniak with less than five minutes played.
However it was United who broke the deadlock after 14 minutes when Dundee goalkeeper Rab Douglas came to collect a Willo Flood free-kick but made no connection, giving Gunning the easiest of tasks to head the ball into the unguarded net at the back post.
The Dark Blues were pressing for an equaliser but were almost caught on the counter when the unmarked Gardyne cut the ball back for John Rankin but the midfielder's acrobatic effort drifted harmlessly wide.
But United doubled their advantage 10 minutes before the break when Rankin threaded a perfect pass into the path of Russell, whose shot looked to be heading wide before a deflection off Italian defender Davide Grassi took the ball past the helpless Douglas.
And Russell extended the United lead with a spectacular strike three minutes later. The forward collected the ball from Barry Douglas before cutting inside and unleashing a venomous shot from 25 yards which flew into the net.
Douglas went some way to redeeming himself with two fine stops after the break, clawing away a Richie Ryan drive before getting down well to keep out a Jon Daly header as United's dominance continued.
Dundee almost pulled a goal back when John Baird picked out Ryan Conroy in the box but the winger's low drive was well saved by Cierzniak before Boyle was introduced for Jim McAlister.
Both teams had to play the final 20 minutes a man short after a tackle between Dundee skipper O'Donnell and Russell sparked a melee between both sets of players and when order had been restored the two players were given their marching orders by referee Calum Murray.