Ryan Holms earned Queen's Park a place in the last 16 of the Homecoming Scottish Cup and a money-spinning trip across the city to Celtic.
Queen's created a hatful of chances in a match they totally dominated from start to finish.
After just five minutes the impressive Paul Cairney found Stevie Nicholas in space, but his 20-yard shot was just wide of the mark.
Ten minutes later Tony Quinn put Ian Watt through but Peterhead defender David Donald made a terrific goal-saving tackle.
Cairney then drew a fabulous save from Polish goalkeeper Michal Kula and just before the interval Paul Harkins reacted first after Kula spilled the ball but Donald was on the line to clear.
Queen's finally got the winner they richly deserved on 51 minutes when Harkins whipped over a cross from the left for Holms to dive between two defenders and send a glancing header past Kula.
Teams
Queens Park Crawford, Ure, Douglas, Sinclair, Brough, Quinn,Holms, Gallacher, Nicholas, Harkins, Watt.
Subs Not Used: Boslem, Ronald, Coakley, McGrogan, Cowie.
Goals: Holms 51.
Peterhead Kula, Donald, Moore (Mann 57), Smith, MacDonald,McVitie, Sharp, Ross (Gunn 59), Bavidge (Kozminski 72),Anderson, McKay.
Subs Not Used: Bagshaw, Jarvie.
Att: 782
Ref: D Somers (Scotland).