Ricky Waugh captained the Black and White youngsters who started off at a fast pace with Lucas McCormick driving deep into the Kilmarnock box in the early stages.
Calum Ferrie was well placed to save the first effort on target in the game on five minutes when Aaron Brown tried his luck from 20 yards with a shot with home striker Innes Cameron heading wide shortly afterwards.
On 11 minutes Queen’s created their first chance when McCormick got free on the left and crossed for McLeish however his header was off target.
Two minutes later McLeish was on target with a curling 20 yard shot with Kieran O’Hara in the home goal diving to his left to save.
On 14 minutes the Ayrshire side took the lead when the experienced Liam Polworth threaded a pass through the red-shirted backline for Cameron to run onto and slot under the advancing Ferrie.
The visitors almost hit-back instantly as good pressing from McCormick won the ball for McLeish in the Killie box with McCormick just an inch away from turning in his team-mate’s reverse pass.
Midway through the half Ferrie dived low to stop a Ben Brannan drive and then leapt up to smother the loose ball.
Rocco Hickey-Fugaccia was causing the home defence plenty of problems with his pace on the right and his dashing run and cross on 30 minutes handed McLeish a glimpse of goal however he was crowded out with Alex Fairlie inches away from goal with a swerving shot from edge of the box when the ball ran loose.
Rather than draw level Queen’s found themselves two down on 30 minutes when Fraser Murray picked out Bobby Wales with a cross and the striker nodded in.
Before the break Ferrie used his knees to stop a Joe Wright header creeping in before making a fine double stop to prevent Wales notching his second with Hickey-Fugaccia creating further danger with another dashing surge forward.
Young Queen’s Park started the second half with a period of pressure on the home goal which only ended when Zach Mauchin was just unable to slide in a clever cross from Waugh at the back post.
Another chance came on 53 minutes when McCormick darted across the pitch with his pass to the unmarked Hickey-Fugaccia unfortunately overhit and running out of play.
Two minutes later a strong Mauchin run was only ended 12 yards from the Kilmarnock goal with Lennon Connolly just out of luck as he tried to hook the bouncing ball home.
On 56 minutes Young Queen’s Park pulled a goal back as Mauchin pushed a ball forward and a dummy allowed MCLEISH to pounce and the striker took his time to pick his spot and shoot home.
Sixty seconds later McLeish was free again however he was fouled 19 yards out with his direct shot from the resultant free-kick flashing over.
Another chance was created just after the hour mark as tenacity from McCormick allowed him to run 30 yards with the ball towards O’Hara’s goal with the shot-stopper able to get down and save the wide-man’s shot.
On 78 minutes the score was tied at 2-2 when a low corner from the left bounced across the six-yard box and ricocheted off a couple of home players before MCLEISH applied a final touch to force the ball over the line.
With seven minutes left a free-kick was cut back to Polworth and his first time shot came off the top of the bar.
On 88 minutes Lewis Reid pushed his side forward and sent substitute Kane Thomson free with Kilmarnock’s Wright sliding in and handling the ball with MCLEISH blasting home the resultant penalty to put Queen’s ahead.
In injury time Ferrie produced an outstanding stop to push out a header from close range with Mauchin breathing a sigh of relief as the ball deflected off him and against a post.
Queen’s Park – Calum Ferrie, Joseph Smith, Zach Mauchin, Alex Bannon, Lewis Reid (Archie Graham 90+4), Ricky Waugh, Rocco Hickey-Fugaccia, Lennon Connolly (Andrew Lind 67), Liam McLeish, Alex Fairlie (Kane Thomson 82), Lucas McCormick (Tyrece McDonnell 67).
Unused substitutes – Liam Russell(G/k), Benjamin Ferguson, Chris Mahon.
Kilmarnock – Kieran O’Hara, Fraser Murray, Aaron Brown, Ethan Brown, Joe Wright, Ethan Mersey, Ben Brannan, Liam Polworth, Innes Cameron, Steven Warnock, Bobby Wales
Unused substitutes – Aidan Glavin, Aaron Quigg, Ruari Ellis, Andrew Miller, Zander Craik, Nathan Smith.