Scottish Cup holders Hearts progressed to the fourth round thanks to a hat-trick from Andrius Velicka.
The striker opened the scoring with a neat lobbed finish and the Lithuanian added a close range shot from a corner kick just before the interval.
Hearts goalkeeper Craig Gordon had to make good saves to deny David Hamilton and Michael Moore.
But Roman Bednar raced through to blast Hearts further ahead and the impressive Velicka clipped in a last-minute shot.
The home team started brightly, pressing the ball enthusiastically and making life difficult for the Premier League side.
But Hearts made the breakthrough after 17 minutes when Velicka ran on to a long ball through the middle from Bruno Aguiar and lobbed advancing keeper Scott Black for his eighth goal of the season.
Stranraer were handed a chance to level after 26 minutes when Neil McCann was caught in possession on the edge of his own box.
The ball fell to Alex Burns but the veteran striker poked a weak effort wide of the target. And a headed flick from Moore allowed Hamilton to produce a sharp shot on the turn which Gordon smartly saved.
But, two minutes before the break, Velicka added a second Hearts goal when he poked home from close range after an Andrew Driver corner was flicked on at the near post.
Stranraer continued to harass their opponents and Hearts failed to find any passing fluency after the interval.
With Hearts not doing anything to trouble the home defence, Moore tried his luck with a powerful shot on the turn from 30 yards but Gordon sprung acrobatically to push the ball away.
Hearts found more room in attack as the game entered the closing stages and substitute Bednar crashed in a shot from 18 yards after running on to a neat pass from Velicka.
Driver and Bednar went close and Saulius Mikoliunas hit the post from long range before Velicka completed his hat-trick in injury time.
The stiker played a neat one-two with Aguiar before stabbing the ball high into the net beyond the prone figure of Black.
Stranraer: Black, Snowdon, McAlpine, Aitken (Walker 89), Wilson, Sharp, Hamilton, Crilly (Wright 51), Moore, Burns (Mullen 45), Ramsay. Subs Not Used: Morrison, Hunter.
Booked: Moore, Snowdon.
Hearts: Gordon, Neilson (Mikoliunas 68), Wallace, Karipidis, Berra, Barasa, Aguiar, McCann (Jonsson 81), Driver, Velicka, Pospisil (Bednar 63). Subs Not Used: Banks, John Armstrong.
Booked: Velicka, McCann. Goals: Velicka 17, 43, Bednar 79, Velicka 90.
Att: 5,100
Ref: E Smith