Falkirk were eliminated from a second competition in 24 hours in a heavy defeat at a freezing Ibrox.
The young Bairns took the lead through Kevin O'Hara before the Glasgow side hit back, and then some.
Jamie Barjonas struck a double in the first half to send Alan Maybury's side behind at the break, and afterwards Zak Rudden scored a penalty, before Billy Gilmour and Jack Thompson added gloss to the result.
The Bairns took the lead when a strike by O'Hara hit Thpompson's outstretched hand in the area. O'Hara took the spot-kick and sent it low to Robbie McCrorie's right, but the keeper beat the ball out. The young striker followed his shot though and tucked away the rebound to give the Bairns the lead.
Rangers hit back six minutes later with a wonder strike from Jamie Barjonas after Thompson had a header rattle the post, and the ball was pushed out of the box and then despatched into the top corner by the midfielder.
He added a second from outside the box by firing past Lewis McMinn in a well-taken second.
It would have been a more severe score line for the Bairns had Cameron Eddie not scrambled a Gilmour effort off the line following a lovely chipped pass from Rudden.
Falkirk were captained by Cameron Blues and included Mark Finlayson who showed up well in the early stages at right-back. Young Mark Stowe was fleet of foot in the first half but the Bairns couldn't inflict further damage on McCrorie's goal.
Alan Maybury's side were still in it at the break but the next goal proved pivotal. It came for Rudden from the spot after Jason Jarvis was adjudged to have fouled inside the box, though the referee had to be informed by his linesman after initially awarding a free-kick.
From then on it was one-way traffic and classy midfielder Gilmour added a fourth for Rangers, while Connor McBride replaced Reis Peggie.
There was no way back for the Bairns and another stunner from outside the area gave Rangers a fifth via Thompson to send Rangers into the semi-finals.