A first-half double from new striker Steven MacLean followed by late goals from Murray Davidson and substitute Liam Craig at McDiarmid Park on Tuesday night sealed the third-round tie for Steve Lomas' side who survived a slight scare when midfielder Michael Keenan levelled for the plucky visitors on the half-hour mark.
Queen's Park were reduced to 10 men when full-back Andrew Robertson was shown a straight red card from referee Kevin Clancy for a late lunge on Gary Miller before Davidson and Craig both struck.
Saints were in fine fettle after back-to-back league wins over Celtic and Ross County but were wary after Third Division leaders Queen's had knocked Dundee out in the previous round.
The visitors arrived dreaming of causing another upset and did not hang about in launching their first attack.
Winger Lawrence Shankland cut inside from the right and let fly with his left foot. Tam Scobbie intercepted to deflect the effort wide for a corner.
Then, at the other end, the home team missed a golden opportunity to go ahead after six minutes.
Tom Scobbie started the move by releasing David Robertson down the left and his cross found MacLean.
With the goal at his mercy, somehow MacLean shot over from just three yards when it seemed easier to score.
But after 17 minutes one-time Rangers prospect MacLean went from sinner to Saint as he bundled the hosts ahead.
Robertson's cross caused panic in the Spiders defence and MacLean was on hand to force the ball home beyond goalkeeper Neil Parry from close-range.
Saints sensed they could kill the tie off early as they piled the pressure on but after 25 minutes they almost paid the price for some sloppy defending.
David McCracken gave away possession midway inside his own half and David Anderson fed the on-rushing Jamie Longworth.
Longworth drove towards goal and saw his 20-yard shot tipped over the bar by Jonny Tuffey for a corner to the part-timers.
But on the half-hour mark, Michael Keenan made it 1-1 with a spectacular 40-yard effort which Tuffey looked to have covered until it flew under his body and went in off the post.
After 36 minutes, Nigel Hasselbaink released Gregory Tade down the right and his cross picked out MacLean who turned the ball over the bar from point-blank range.
But after 39 minutes MacLean put Saints back in front with a fine 14-yard finish after Davidson had set him up with a perfect pass.
Early in the second half, Steven Anderson should have made it 3-1 but his free header from Scobbie's flighted free-kick hit the bar and went over.
With 16 minutes left, Queen's defender Robertson was rightly sent off for an ugly challenge on Saints full-back Miller as the part-timers' hopes of a comeback disappeared.
Davidson made it 3-1 soon afterwards with a cool 18-yard finish from Tade's cut-back, before substitute Craig's sweet volley from inside the penalty area sent Saints comfortably, in the end, through to the last eight.