Steven Whittaker converted twice from the spot before failing to claim his hat-trick by missing the third spot-kick, as Mihael Kovacevic, Sean Dillon and Morgaro Gomis were all dismissed for denying clear goalscoring opportunities.
Nikica Jelavic and Kyle Lafferty were also on target to ensure the SPL champions head into this weekend’s Old Firm derby at the summit, regardless of the outcome of Celtic’s game at Kilmarnock tomorrow.
Rangers made three changes from the side which beat St Mirren at the weekend for the clash at Tannadice.
Madjid Bougherra was back after five games on the sidelines through injury and started along with Kyle Hutton and Steven Naismith, who returned from suspension. El-Hadji Diouf, Jamie Ness and Richard Foster dropped out of the side.
United also made three changes. Kovacevic made his first start of the season after a cruciate ligament injury and David Robertson and Scott Robertson also started, while Paul Dixon was suspended and Craig Conway and Prince Buaben were named on the bench.
Rangers could have applied some pressure when Maurice Edu won a free-kick and Steven Davis delivered into the packed goal-mouth only for goalkeeper Dusan Pernis to gather before anyone could connect.
Naismith then collected from Jelavic and had the ball in the back of the net but he handled in the build-up and the effort was disallowed.
The match exploded when Lafferty drew a save from Pernis, before Kovacevic fouled Naismith in the box and referee Willie Collum pointed to the spot.
The full-back was then dismissed in similar circumstances to Aberdeen’s Andrew Considine, who was sent off by the same match official in the Scottish Cup at the weekend.
Whittaker stepped up for the spot-kick and sent a low shot into the bottom corner after 22 minutes as he converted from 12 yards for the second time in as many games.
United immediately made a change and David Robertson was withdrawn for Barry Douglas, before Jelavic dragged a shot wide as Rangers attempted to double their advantage.
Hutton picked up a booking and was replaced just three minutes later by Diouf, as Rangers made their first switch with 34 minutes gone.
Naismith then saw his header saved by Pernis, before Lafferty was again denied by a decent block from the goalkeeper just before the break.
Recent matches had been nervous, hard-fought affairs for Rangers and they set about trying to add to their tally in the second half.
A foul by Douglas on Naismith provided an opportunity to put the United defence under pressure and Diouf whipped a dangerous ball into the box but Jelavic nodded just over.
The Croatian striker squandered another chance in front of goal moments later, this time meeting a decent ball from Davis and failing to keep the volley down and on target.
Rangers doubled their lead after an hour when the referee pointed to the penalty spot and reached for his red card for the second time in the game, as Dillon was sent packing for tripping Jelavic. Again, Whittaker held his nerve to score.
There appeared to be no way back for United when Rangers grabbed their third of the night after 72 minutes.
Davis forced a decent block from Pernis but the goalkeeper spilled into the path of Jelavic who netted the rebound from the tightest of angles to claim his fifth goal in his last seven matches.
There was further drama to come when Gomis was sent off for a push on Diouf and Rangers were handed their third penalty with 12 minutes to go. However, Whittaker failed to find the net on this occasion, blasting high and wide.
Lafferty then sealed the victory after 82 minutes when he controlled a neat lay-off from substitute David Healy before rifling home from 25 yards.