Although the visitors took the lead, the Bairns forced extra-time through goals from Rory Loy and Craig Sibbald.
Blair Alston then proved the hero for the home side with a dramatic 118th-minute winner.
Motherwell loanee Robert McHugh, who netted both goals in Queens’ 2-1 first-leg success, was on target again in the 36th minute with a close-range volley.
But Falkirk levelled eight minutes into the second half when Loy swept the ball past Zander Clark following fine work by Jonathan Flynn.
The home side moved in front in the second leg and tied the aggregate score at 3-3 with 20 minutes left as Sibbald nodded past Clark to force extra-time.
Loy had a golden chance to put Falkirk further in front in the first period of extra time when he was one-on-one with Clark, who denied him.
But just as the game looked to be heading to penalties, Alston was presented with a tap-in to win the contest after good work from Scott Shepherd.
Falkirk will now take on Hamilton in the semi-final, with the winner progressing to a final date with Hibernian.