Attendance 10,654
In a typical cup-tie played before a full house in the old ground, the First Division club gave a good account of themselves and at times threatened an upset.
Rangers made no fewer than seven changes from Saturday with Neil Alexander, Danny Wilson, DaMarcus Beasley, Jordan McMillan, Kevin Thomson, John Fleck and Nacho Novo replacing Allan McGregor, David Weir, Lee McCulloch, Steve Davis, Steven Naismith, Kenny Miller and Kris Boyd. For Wilson and McMillan it was a first start at this level.
With Sasa Papac as Captain, the visitors came close to opening the scoring in fourteen minutes when Fleck's low cross was met by Novo ghosting in at the far post - the little Spaniard's shot being turned wide by home goalkeeper Tony Bullock.
The opener was merely delayed - Steven Whittaker heading home from Fleck's resultant corner.
Ten minutes later Dundee had a chance to equalise when Craig Forsyth was through on goal only to be blocked by the outrushing Alexander. There were claims for a penalty, but Referee Dougie McDonald waved play on.
Fleck was creating problems down the left, and from one penetrating run in 27 minutes he found Novo with his cutback only for the Spaniard's weak shot to be held by Bullock.
The scores were levelled two minutes later when Leigh Griffiths' 22-yard free-kick exploded into the net behind Alexander.
Griffiths was causing problems for the Ibrox men - cutting in from the right in 35 minutes, his angled drive was beaten away by Alexander.
It was the First Division side who made all the running at the start of the second-half - Griffiths' mazy run down the right in 53 minutes ended with a low shot that Alexander parried, then three minutes later Gary Harkins saw his twenty-yard free-kick held by Alexander.
Remarkably the goal when it arrived came at the other end in what was almost Rangers' first incursion into home territory of the half in 57 minutes - Whittaker's cross being headed bizarrely into his own net by Gary MacKenzie.
The tie had turned irrevocably and fortuitously in the visitors' favour, although Dundee substitute Colin McMenamin did threaten with a header from a Harkins cross that flew over the bar in 75 minutes.
Rangers sealed victory with six minutes remaining when in a lightning break Thomson released Novo on the halfway line, the Spaniard streaking away down the loft before finding the unmarked Fleck who stroked the ball home.
Afterwards Manager Walter Smith commented:
"We always knew this was going to be a tough game. Dundee played well, but I was happy with our commitment and the fact that we put recent matters out of our minds."
DUNDEE Bullock; Paton, Malone; Forsyth, MacKenzie, Lauchlan; McHale, Kerr (Hart 78), Griffiths (McMenamin 71), Higgins (Clarke 68), Harkins
UNUSED SUBS Soutar, Casement
RANGERS McGregor; Whittaker, Wilson, Papac, Smith; Beasley (Naismith 74), McMillan, Thomson, Fleck; Novo, Lafferty
UNUSED SUBS McGregor, Davis, Boyd, Miller
REFEREE Dougie McDonald