Stuart Armstrong finished decisively for the opening goal, which came after the Ibrox side spurned a clear chance.
Gary Mackay-Steven's second came via a deflection and the Rangers goalkeeper Steve Simonsen made an error for Nadir Ciftci's third.
despite a first-half Steven Smith free-kick reducing the deficit Rangers were unable to truly threaten United's passage to a final where they will meet either Aberdeen or St Johnstone on 17 May.
The game initially had to accept second billing, since the competition was between the two sets of supporters and their goading of each other. The noise spilled from the Ibrox stands as if oblivious to events on the pitch.
Dundee fans were angered by having to play the semi at rangers 'home' ground. Earlier Scottish FA, Chief executive Stewart Regan had defended the decision to announce venues for the Scottish Cup semi-finals early in the season.
The SFA confirmed in October that Rangers' Ibrox Stadium would host both matches due to Hampden's unavailability.
With a place in the final secured you won't hear many complaints after the event and the SFA's decison is vindicated.