There was an early scare for Stevie Farrell’s side when Adam Cummins failed to cut out a Scott McCrory-Irving cross , but Gregor Woods at the back post could only stab the ball straight at Greg Fleming.
The home side took the lead on eight minutes, however.
Ayrton Sonkur’s header from an Andy Stirling corner went straight up in the air, but as Spartans hesitated, waiting for the ball to drop, Darryl Duffy reacted quickest to race in quickly and power a header past Matt Baines.
Fleming saved well from Scott Maxwell at his near post to preserve Stranraer’s lead, which they then doubled as the visitors’ defence allowed Stirling’s delivery to travel all the way across to the far post, where Ruari Paton was waiting to knock it past Baines.
Stranraer added a third when Jamie Hammill’s headed flick from another Stirling corner found its way through to Cummins and his shot made it into the net despite the despairing attempts of a defender to keep it out.
Farrell made a change five minutes from the break, taking off Duffy as a precaution after he picked up a knock and sending on youngster Matty Yates.
Farrell has high hopes for the former Rangers youngster after he impressed as a sub against Albion Rovers, and he didn’t disappoint.
Midway through the second half, Hammill picked him out with a cross from the right and he cleverly guided a header over Baines to put Stranraer four up.
It was the 18-year-old’s first senior goal and he didn’t have long to wait for his second.
Three minutes from time, Stirling again broke on the right and laid the ball on a plate for Yates to knock home.
Stranraer have been drawn away to Kelty Hearts in the second round.