The young Spiders lost two goals in each half with penalty decisions denying them the chance to get a foothold in the game at key times.
Aaron Healy was given a run-out in the tie and he combined with Gregor Nicol to create danger after just two minutes however the hosts got players back to cover.
With 11 minutes played Rangers threatened when Robbie Ure slipped a pass through to Anthony Weston however Callan McKenna spread himself to save with the shot-stopper doing well again two minutes later when Cole McKinnon drilled in a shot after a Ure knockdown.
McKenna then found himself in the right place at the right time to clutch a shot from Weston after the visitors had failed to clear their lines.
Midway through the half a fine cross-field move involving Nicol and Calum Biggar freed Healy on the left with the winger’s cross flashing across the Rangers goalmouth before being cleared.
On 25 minutes Rangers took the lead when Ure turned on the spot and sent a right wing cross past McKenna.
Queen’s were still trying to get to grips with being behind when Thompson Ishaka tumbled to the ground on 30 minutes with referee Murdoch pointing to the spot. It looked a soft award however that did not trouble Ishaka who drilled home from the spot.
Four minutes later a second penalty was awarded after a challenge on Ishaka however this time McKenna stuck up a hand and saved the striker’s effort from 12 yards.
Before the break Healy caused danger on the left again however his deft cross found no takers.
Eight minutes into the restart Cameron Bruce kicked clear after Ishaka had slipped the ball past McKenna with Queen’s Park giving Rangers a reminder they were still in the game on 55 minutes with a fine drive from Gregor Nicol that Lewis Budinauckas needed two attempts to hold.
Just after the hour mark it looked like the visitors might get a route back into the game from the penalty spot when Liam McLeish appeared to be nudged off the ball as he rose to win a header.
No whistle came and Rangers broke up to the other end and McKenna had to leap to his left to push away a shot from Zak Lovelace.
With 19 minutes left substitute Tyrece McDonnell darted down the left to get into a good position only for the Rangers defence to regroup and crowd him out as he looked for team-mates.
On 83 minutes it became 3-0 when Ure scored a carbon copy of his first goal by sweeping in a right wing cross and as the hosts increased the pressure McKenna denied James Graham a goal.
Five minutes later Graham released Connor Allan on the left and his back post cross was thumped in by Lovelace to complete the scoring.
Queen’s Park: Callan McKenna, Cameron Bruce, Calum Biggar, Alex Fairlie (Liam McQuaid 60’), Aaron Healy (Tyrece McDonnell 60’), Gregor Nicol (Joseph Smith 80), Zach Mauchin, Kane Thomson (Archie Graham 46’), Lewis Reid, Andrew Lind (Lennon Connolly 46’), Liam McLeish.
Subs not used: Liam Russell (G/k).
Rangers: Lewis Budinauckas, Robbie Fraser, Connor Allan, Lewis MacKinnon, Bailey Rice (Charley Lindsay 80’), Thompson Ishaka (James Graham 66’), Cole McKinnon (Mackenzie Strachan 80’), Robbie Ure, Aaron Lyall (Zakariya Lovelace 60’), Anthony Weston (Kelsey Ewen 80’).
Subs not used: Jacob Pazikas, Kevin Ciubotaru.