The League One champions were ruthlessly punished for defensive lapses as Stuart Armstrong, Gary Mackay-Steven and Nadir Ciftci took full advantage.
United opened the scoring on 26 minutes when Gavin Gunning sent a header from 40 yards over the advancing Rangers rearguard and Armstrong had time to let the ball come down and bounce again before sending a cushioned shot into the net off the inside of the post.
Rangers were punished again when Ricky Foster hit an attempted clearance straight to Ciftci before Ryan Gauld nutmegged the Rangers defender and cut-back a pass for Armstrong to set up Mackay-Steven to scramble the ball past Steve Simonsen.
Stevie Smith pulled a goal back for Rangers with a dipping free-kick from 25 yards which squirmed through Rado Cierzniak’s hands three minutes before the break.
But Cifci settled the outcome seven minutes from time when Simonsen fluffed a clearance straight to the striker who walked the ball past the Rangers keeper and tapped it into the net.
It was a spirited performance by the third-tier side which at times threatened to boil over as Lee McCulloch wrestled Mark Wilson round the neck and chucked the United defender to the deck without any punishment, while Foster was lucky to escape without a red card for a horrible challenge on Armstrong which did not even receive a booking.
Rangers started brightly and Andrew Robertson got in a timely block to thwart Bilel Mohsni on the six-yard line before John Rankin turned the ball behind for a corner after only three minutes.
And the Glasgow team had an even better chance when Dean Shiels got in behind the United defence but screwed his shot well wide from 15 yards.
Nicky Law later had a cross which dropped on to the upright as United struggled to get out of third gear.
And the United’s defence switched off again when Fraser Aird beat the offside trap and laid the ball back for Jon Daly, but the big Irishman crashed his shot high and wide from 12 yards.
A minute after taking the lead Rangers where opened up again when Gauld sent a first time pass to Ciftci on the right side of the penalty area and Simonsen got down to push the ball wide from his powerful low drive.
United were denied a penalty though when Bilel Mohsni barged Ciftci to the deck as the striker prepared to round Simonsen, but referee Bobby Madden waved play on.
The Scottish Premiership side looked comfortable for the majority of the second-half but were almost undone by another Smith free-kick from 20 yards which the deflected off Gunning but Cierzniak atoned for his earlier blunder by clawing the ball round the post.
Three minutes later Ciftci was inches away from turning in a cross from Keith Watson as United opened up Rangers on the right flank, and Mackay-Steven fizzed a shot inches wide in the closing moments.