A goal from substitute Myles Hippolyte right at the end of extra time took Livingston through, but this was an uninspiring match between two sides who cancelled each other out for long spells.
It might have been a different story if the deadlock had been broken early but Gary Glen, who earned a place in the Livingston starting XI after making an impact from the bench at the weekend, miscued in a congested Queens box.
A little more composure in possession would have served either side well but the lack of it saw another chance spurned by Livingston when Jordan White lashed the ball high over the bar. There was an element of haste, too, when John Baird headed Queens' first attempt straight at Darren Jamieson from Iain Russell's cross.
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A series of niggling fouls militated against any sustained flow to the play but one of them almost brought the opener for the hosts when Burton O'Brien sent a 25-yard free kick inches over the bar.
Queens goalkeeper Zander Clark pulled off a great save to keep his side level as half-time approached, diving full length to save Keaghan Jacob's netbound effort.
The hosts showed some urgency after the break when O'Brien burst forward and sent an angled shot narrowly wide before Michael McKenna's effort was blocked on the edge of the box. The same player was thwarted by Clark's reflexes before the Queens goalkeeper again came to the rescue on the hour mark with an athletic double save to deny White and Jacobs.
White also went close with a header which flashed over the bar while Jason Talbot volleyed over as Livingston continued to force the pace. The overtime might have been averted had Queens substitute Derek Lyle managed to get a late shot on target.
As the game drifted towards a penalty shoot-out, Mark Burchill found Hippolyte in the box and he showed the composure which had been lacking earlier to finally beat Clark.