On what was at times a tough night Walter Smith’s depleted team showed great character to overcome Jocky Scott’s men and put the club’s recent off-the-field issues to the side.
The gaffer named three youth stars in his starting line-up and Jordan McMillan, Danny Wilson and John Fleck were impressive and assured throughout.
A first-half header from Steven Whittaker put Rangers on their way but Leigh Griffiths eqaulised with a fine free-kick before the break.
In the second-half Dundee were again dangerous but an own goal from Gary McKenzie and a fine third from Fleck at the death made it a great night for the travelling Bears,
Smith confirmed in yesterday’s pre-match press conference that Wilson would be handed his competitive first-team debut.
And the 17-year-old started in the heart of defence alongside Sasa Papac who wore the captain’s armband with David Weir rested.
Fleck, 18, was also in the team but the big news was that McMillan joined fit-again Kevin Thomson in the centre of midfield.
The 21-year-old Glaswegian went on loan to Hamilton Accies last season having come through the youth ranks at the club he supported as a boy.
And his display tonight will give Smith another option as Rangers aim to fight on all fronts this season.
With Madjid Bougherra, Mo Edu, Kirk Broadfoot, Lee McCulloch, Andrew Little and Pedro Mendes all out through injury Smith’s options for tonight’s game were limited.
And with massive SPL and Champions League games against Dundee United and Unirea Urziceni coming up the manager was always going to shuffle his pack for this trip to Tayside.
In goals Neil Alexander returns and his goal will be protected by a backline consisting of Steven Whittaker, Papac, Wilson and Stevie Smith.
DaMarcus Beasley makes his first appearance of the season on the right wing with Fleck on the left and Thomson and McMillan in the middle.
Up front the little and large combination of Nacho Novo and Kyle Lafferty were asked to lead the line and both worked hard for the cause on this trip to the City of Discovery.
Dundee, however, showed their intent early on and showed a level of confidence expected for a team sitting at the top of the First Division and looking good for promotion.
Former Partick Thistle winger Gary Harkins, in particular, looked to unlock the Gers’ backline at every given opportunity and he proved a handful for Steven Smith.
But Rangers also enjoyed a positive start and they took the lead on 14 minutes althoughJocky Scott must have been tearing his hair out at his defence.
Only two minutes before his side conceded they were given a warning when Fleck took an inswinging corner and Steven Whittaker shot wide having darted to the front post.
In an almost identical situation Fleck again looked for the right-back and he was given a second opportunity to find the net.
This time he made no mistake and his downward header from close range beat Tony Bullock.
Immediately after the opening goal there was a moment of controversy at the other end when Craig Forsyth was played through on goal and he appeared to be brought down by Alexander as he tried to round the keeper.
Dougie McDonald let play continue to the disgust of the home bench who clearly thought a spot-kick should have been awarded.
Rangers, however, remained in the ascendancy and went in search of a second.
And they should have got it when Fleck did brilliantly to skin full-back Eric Paton. His cut back found Novo but his shot on the spin lacked the power and accuracy to beat the Dees keeper.
This was a turning point in the first-half as the home side went up the park and scored before the half hour mark.
The ref awarded a foul 25 yards from goal and the impressive Griffiths curled a beautiful free-kick past the helpless Alexander.
The striker then had a chance to put his team in the lead before the interval but his near post effort was palmed away this time by the keeper.
At the start of the second period Dundee came flying out of the traps and Rangers had to dig deep to keep the game level.
In the first minute Forsyth made another driving run in to the box but his touch was poor and the ball went out for a corner.
Griffiths then showed his pace and skill again when he beat Smith but his powerful shot was parried by Alexander who was again on top form.
In an intense six minute period of pressure Paton then blasted a first-time volley from a corner high and wide of goal.
At this stage it looked certain that Rangers needed a change but they were to get a huge slice of fortune before the hour mark when they regained their goal advantage.
Whittaker delivered a floated cross from the right that looked to be going harmlessly through to Bullock.
But in a moment of madness former Murray Park defender Gary McKenzie leaped to head the ball away and glanced it in to the net with his keeper stranded.
The goal stunned the home support and there were further anxious moments for them when former Dens favourite Novo fired a free-kick which went narrowly wide of target.
But this game was not over by a long shot and when the dangerous Harkins crossed from the left Alexander watched carefully as substitute Colin McMenamin headed inches over.
With five minutes remaining, however, Rangers sealed their passage through the semi-finals with an excellent counter attacking move.
It ended with Novo finding Fleck in acres of space in the box and the youngster, arguably Rangers’ best player on the night, slotted it home with aplomb.
This was far from an easy 90 minutes for Walter Smith’s side but they can travel back to Glasgow with their heads held high and McMillan, Wilson and Fleck can be proud of their performances.
Man of the Match: John Fleck – The youngster ‘s movement and use of the ball was excellent and he thoroughly deserved to get on the scoresheet.