Velicka breezed past Lee Sharp and lobbed home keeper Scott Black as the Jambos got their show on the road in the 17th minute.
But the second division men battled back and both David Hamilton and Alex Burns might have dragged the tie level.
Two minutes before the interval Stranraer were hit by the sucker-punch when Velicka headed home after Andy Driver's corner was knocked on into his path by Michal Pospisil.
Gerry Britton's home side sent three men upfront for the second half and it almost paid off in spectacular fashion when Michael Moore's 25-yard left-foot shot had to be tipped to safety by Craig Gordon.
But Stranraer tired and substitute Bednar made it three with a powerful 79th minute rising drive and then Saulius Mikoliunas cracked a shot off the woodwork.
Velicka wrapped up his hat-trick when he squeezed a clever goal home at the post deep into second-half injury-time.
Hearts avoided what might have been a banana skin but they will certainly know they had been in a game despite the rather one-sided scoreline.