The Lesser Hampden encounter, coming just under 24 hours after the clubs’ first teams clashed at Ibrox, was not a memorable meeting.
However a goals either side of half time were enough to see Billy Kirkwood’s team through as Luca Gasparotto, Calum Gallagher, Jamie Burrows and Craig Halkett got the strikes that mattered.
In truth the first period produced few real chances as Greg Pascazio headed over from a corner and blocked the home side’s only really chance at the other end.
It was a mix up in the home defence that opened things up for the young Gers as Canadian defender Gasparotto was presented with an open goal five minutes before the break.
He made no mistake in hooking the ball home and Rangers might then have extended their lead before the interval.
Gallagher would have been better served squaring the ball for Tom Walsh when he shot well over the top, and Andy Murdoch might have finished after appearing to get round the keeper on the stroke of half time.
As it was the half ended at 1-0 to the visitors and there was a scare for them early in the second half when Queens Park had the ball in the net.
The half was only eight minutes old when Blair Currie produced a brilliant stop only to see the rebound lashed high in to the net.
Thankfully for Currie and his defence the linesman had his flag up and the goal was ruled out for offside.
A change in shape on the hour mark with the introduction of Craig Halkett for Tom Walsh saw Halkett join half time sub Burrows up front.
That moved Gallagher out to the wing but he showed his striking instinct to get in to the box and latch on to the ball when Halkett was denied.
The prolific front man showed again why he has been so highly praised by Kirkwood and Reserve coach Tommy Wilson this year with the coolest of finishes under the unsighted keeper.
That effectively ended the game as a contest and Burrows’ late contribution, a neat finish inside the box and Halkett's simple tap in added a little gloss to the score line for Kirky’s kids.
Attention now turns to the draw as Rangers look to continue progress in the Cup with a young squad.
Rangers: Currie; Sinnamon, Gasparotto, Pascazio, Clarke; Murdoch, Wiktorski, Ramsay (Wilson 87); Telfer (Burrows 46), Gallagher, Walsh (Halkett 62).
Subs: McRorie, Urquhart.