The Gers took the lead just six minutes into the game when Steven Whittaker slotted home from the edge of the box. Sone Aluko then put the match beyond reach with two fine individual efforts before Jamie Ness and Alejandro Bedoya put the gloss on a comfortable victory.
Ally McCoist's side made a fast start to proceedings when Whittaker finished after some neat link up play from Salim Kerkar and Rhys McCabe.
United came close to equalising moments later when Allan McGregor denied Gary Mackay-Steven from close range.
|18/1||Scotland are 18/1 with Sportingbet to win Group A and qualify for the World Cup in Brazil.|
The visitors were made to pay for this missed opportunity when Sonie Aluko doubled the lead with a superb individual effort, the striker racing past a trio of Dundee defenders before cooly placing the ball past Dusan Pernis.
Shortly after, Aluko repeated the feat, firing across goal and into the bottom corner after evading the defence, to put the result beyond doubt just 20 minutes into the game.
Peter Houston's men only offered sporadic resistance, and Rangers continued in the ascendancy after the half time interval when Aluko forced a good near post stop from Pernis.
The Slovakian keeper could do little to halt the Rangers onslaught as young gun Ness combined well with the outstanding Aluko on the edge of the area before calmly passing the ball past a stricken Pernis.
Dundee's quest to find some solace in the form of a consolation goal proved to be a thankless one as they were continually halted by McGregor, who was magnificent in the Rangers goal. The visitors' best chance fell to striker Johnny Russell, whose powerful close range effort was well blocked by the Scotland international.
United's frustration will have been compounded by the clinical nature of the hosts, as the Glasgow side added the fifth and final goal of the night when Bedoya latched onto a well placed through ball from Maurice Edu to slide the ball in the net and cap a marvellous night for the financially troubled club.
The result leaves Rangers sure of second place, as they sit eight points clear of third placed Motherwell while United are fourth, just three points ahead of St Johnstone in the battle for the final two European spots.