The Irish Cup winners built on their first-leg lead with Jordan Forsythe scoring an early goal in the Faroe Islands before Paul Heatley’s strike.
Hugin Samuelsen pulled one back with a back-post finish before the break and Lukasz Cieslewicz stabbed in to level.
Heatley calmly slotted home the winner and a visit to Molineux awaits.
Crusaders travel to take on Wolves next Thursday in the first leg of the second-round qualifier.
The visitors made the perfect start at a foggy Gundadalur Stadium as Forsythe smacked a six-yard free-kick high into the net in the third minute after keeper Hans Jorgensen picked up a backpass.
Arni Frederiksberg lobbed wide a minute later at the other before the Crues pressed for a second goal with Ross Clarke dangerous on the right.
Heatley doubled the lead with a low 25-yard shot across Jorgensen with the ball sneaking just inside the post.
Torshavn grabbed their first goal of the tie eight minutes before half-time as Samuelsen met a cross from the left to tuck home from close range.
Samuelsen produced a fine double save to deny Heatley and Forsythe early in the second half before the hosts drew level on 50 minutes.
Stefan Radosavljevic picked out Cieslewicz with the cut-back and a weak effort managed to creep into the corner of the Crues net.
Heatley eased any Crusaders nerves in the 68th minute with the winger sliding the ball beyond Jorgensen from 16 yards.
- 3Tór NaesSubstituted forSamuelsenat 83′minutes
- 15SamuelsenSubstituted forAgnarssonat 72′minutes
- 13JacobsenSubstituted forPrzybylskiat 62′minutes
- 1O’NeillBooked at 45mins
- 12CaddellSubstituted forHaleat 65′minutes
- 3HegartyBooked at 5mins
- 14ForsytheBooked at 78minsSubstituted forThompsonat 86′minutes
- 18OwensSubstituted forCushleyat 83′minutes
- 22HeatleyBooked at 90mins
- Milovan Milacic