The strike partnership of Johnny Russell and Jon Daly link up to give new Tannadice boss Jackie McNamara the perfect start, securing a place in the last eight of the competition
The Third Division leaders ended the tie with nine men on the park as both Kal Naismith and Ian Black were shown red cards late on.
New Tannadice boss Jackie McNamara stuck with the same starting line-up that drew at home to Ross County in Peter Houston’s last game in charge on Saturday.
Meanwhile, opposite number Ally McCoist made two changes from the side that failed to beat Montrose last week, with Chris Hegarty and Lewis MacLeod dropping out in favour of Sebastian Faure and Black.
The Ibrox side got off to the worst possible start as the defensive lapses that have riddled Rangers' performances this season were cruelly exposed within the first 15 seconds, as Scotland call-up Lee Wallace failed to deal with Daly, letting the Irishman find Russell and he fired a strike under Neil Alexander to claim the advantage.
Rangers could have levelled straight away as David Templeton found Andrew Little, but his strike from just inside the area sailed wide of Radoslaw Cierzniak’s left-hand post.
The home side should have doubled their lead just after the quarter of an hour mark as Daly once again rose above the Rangers defence to meet a Barry Douglas free kick and nod it on to Russell, but his strike from an acute angle cannoned off the bar and away to safety.
Despite Rangers getting back into the game, it was Dundee United who had the next chance as Daly got onto the end of a Willo Flood cross, six yards from goal, but his effort went straight into the chest of Alexander.
The defensive problems that McCoist’s side have been experiencing this season once again came back to haunt them as Emilson Cribari allowed Daly to drift past him and get a free header from a Flood free kick, which the Irishman dispatched past Alexander to double the lead.
Rangers were weakened at the start of the second half as Francisco Sandaza was forced off with an injury at the break which forced Ibrox gaffer McCoist to call upon youngster Naismith to try and get them back in the tie.
They did start to gain a foothold on the game, but could rarely muster a chance on target, with the best of those coming the way of Black, as he rifled a strike from just outside the box, but the ball curled wide of goal.
The path to the quarter-finals was secured when Daly and Russell linked-up once again, with the Terrors skipper finding his 22-year-old strike partner, who got in behind Wallace to get his second.
Rangers' day got even worse as Naismith made a ridiculous challenge on Flood that earned him a straight red card and he was soon joined by Black who received a second yellow for a challenge on Rankin as the Ibrox side went out of the competition.