It all appeared to be going swimmingly for the young Gers when Cole McKinnon nodded the away side ahead in a first half in which they were firmly in control of both the ball and the creation of chances.
However, the contest pivoted on the decision from referee Peter Stuart to show Clinton Nsiala a straight red card for an off-the-ball incident ten minutes from the interval.
That swung the momentum in Queen’s Park favour, who took advantage in the second period through goals from Dom Thomas, Jack Turner and a Kristian Webster own goal.
The result brings to an end a positive cup run for the young Gers, who will certainly benefit from the experience of tonight and the manner of their performance.
Rangers showed immediate endeavour as they sought an avenue through Queen’s Park in the early exchanges, going close when Bailey Rice’s strike was deflected wide after promising play on the left-hand side from the in-form Findlay Curtis.
The deadlock ought to have been broken two minutes later when the Light Blues sprung forward in a quick transition. This time, Curtis released Chris Eadie beyond the defence but, with only the goalkeeper to beat, the 17-year-old appeared to overrun the ball and Jack Wils was able to smother.
The away side were not so much knocking on the door as taking it off its hinges when, moments later, Archie Stevens fired a dangerous delivery across the face of goal which whizzed beyond everyone in blue.
Stevens then fired over himself after neat combination play on the edge of the danger area, before Rangers eventually got their rewards for their early persistence.
Again, the Gers were able to advance possession into the final third, with Nsio taking the initiative and floating a lovely delivery into the penalty area where Cole McKinnon arrived in space to nod into the bottom corner.
One very nearly became two for the Light Blues – and it would have been no more than they merited – when Robbie Fraser skipped inside and curled a dipping effort which cannoned off the inside of the far post.
Mason Munn was called into action for the first time on the half-hour mark when Jack Turner unleashed a dangerous effort from 30 yards out with prompted a stop at full stretch from the Northern Irishman.
The game then had its defining moment with ten minutes remaining of the first half when Clinton Nsiala was shown a straight red card for what appeared an off-the-ball incident.
It meant that the territory now belonged to the Spiders, who almost immediately capitalised when a cross from the right was swept goalwards by Sean Welsh but wide of the Rangers target.
The half-time whistle was a welcome reprieve for the Gers, but they were pegged back just 90 seconds after the interval when Queen’s Park reduced the arrears through a pinpoint Dom Thomas strike in off the post.
Despite the numerical disadvantage, Rangers continued to play their attacking brand of football as Fraser was only inches away from picking out McKinnon with a slide rule pass from the left-hand side.
The hosts, however, turned the game on its head moments later when a wicked cross from Rocco Hickey-Fugaccia befuddled the Gers defence and Turner arrived to slam home first-time beyond Munn.
David McCallum turned to his substitutes bench for inspiration, introducing 17-year-old Callum Burnside who has impressed at Under-18s level this term.
A one-goal deficit meant the Light Blues always had a chance, continuing to play their way through an aggressive press from the Spiders and sticking to their principles.
But the game was ultimately put beyond them with 22 minutes remaining when Zander Hutton got into a mix-up with Webster and the Gers full-back whacked a clearance against the captain which flew into his own net.
Rangers continued to show endeavour in abundance, with Fraser and Curtis going close, but it wasn’t to be as Queen’s Park booked their place in the last four at the expense of the young Gers.
QUEEN’S PARK: Wils, Mauchin, Scott, Welsh, Thomas (Longridge, 85), MacGregor (Hinds, 60), Tizzard, Rudden (McLeish, 68), Turner (Thomson, 85), Murray, Hickey-Fugaccia (Duncan, 60)
Subs not used: Ferrie, Thompson, Kerr, Fieldson
RANGERS: Munn, Hutton, Webster, Nsiala, Fraser, Rice, McKinnon, Nsio (Burnside, 60), Stevens (Gentles, 68), Curtis, Eadie (Grant, 36)
Subs not used: McGuire, Wyllie, Campbell, McClure, McCallion, Stewart