Harry Forrester struck for the third game running to put the Light Blues ahead after just 13 seconds and from there Mark Warburton's men never looked back.
A close-range header from Jason Holt two minutes after half-time, Andy Halliday's free-kick and a late solo effort from skipper Lee Wallace sealed a stunning victory.
Without cup-tied duo Michael O'Halloran and Billy King and injured top scorer Martyn Waghorn, Gers hoped their front three of Kenny Miller, Barrie McKay and Forrester would be strong enough to shoot-down top-flight opposition for the second round running.
Dundee also went with a bold attacking triumvirate as Kane Hemmings, Greg Stewart and Gary Harkins returned after Hartley started them on the bench for Wednesday's goalless draw at Celtic Park.
Controversial comments made by both Harkins and Darren O'Dea on the Light Blues in recent months ensured both were booed roundly by the Ibrox faithful.
Ex-Celt O'Dea reckoned Rangers should have been stripped of the trophies won during the EBT-era, while Harkins sparked an outcry after joking the post-liquidation Gers were a new club.
Rangers' Jason Holt (2nd R) celebrates scoring their second goal against Dundee with Kenny Miller.
Image credit: Reuters
But it was the home support who had the last laugh as their side struck early.
Rangers conceded possession from kick-off. However, Dee left-back Kevin Holt dallied on the ball and was robbed by Forrester, who darted straight for goal before slamming past Scott Bain at his near post.
Ibrox roared with delight and there was almost more to celebrate soon after as Bain denied Tavernier's shot, while Kevin Holt headed against his own post before it bounced clear.
Harkins was jeered as he tugged Dundee's first effort wide but Gers were soon flooding forward again, with O'Dea well placed to cut out Miller's dangerous cross.
Bain pushed away Tavernier's free-kick but for all that Rangers' one-touch passing was a joy to watch, Warburton would have been concerned their half-time lead was not bigger.
Like the first period, though, his team came out flying and got the goal their dominance deserved just 90 seconds after the restart.
Forrester this time turned creator, dinking a lofted pass into the chest of McKay, who knocked it down first-time into Miller's path. The former Scotland striker's angle was too tight to shoot so he calmly steered it back across goal for the onrushing Holt to nod home.
Rangers' Andrew Halliday (3L) celebrates scoring their third goal.
Image credit: Reuters
Gers were then out of sight after 55 minutues. Paul McGinn's foul on Miller was punished with a free-kick just outside Dundee's box and Halliday made it count, sweeping swept a low strike past Bain.
Wallace then underlined Rangers' emphatic victory with seven minutes left as he danced past three challenges before stroking beyond the exposed keeper.
Elsewhere, a dramatic last-minute winner from Billy McKay gave 10-man Dundee United victory in an action-packed Scottish Cup quarter-final against Ross County at Global Energy Stadium.
Liam Boyce gave County the lead midway through the first half but Henry Anier levelled just before the hour mark.
Billy McKay had the final say
Image credit: PA Sport
But when United went down to 10 men on the hour mark when Mark Durnan was sent off and Brian Graham put the home side ahead from the penalty spot there looked as if there was only going to be one winner.
But Anier scored his second goal of the tie to level the scores at 2-2 and McKay struck at the death to see United through to the semis.