Crusaders produced magnificent performance beating FK Ekranas 3-1 at Seaview while Glenavon put up a gallant effort before losing 3-0 to Hafnarfjordur in Iceland, all the goals coming in the last eight minutes.
League of Ireland side Derry City also put one foot in the next round with an emphatic 4-0 win over Aberystwyth Town at the Brandywell.
Sammy Morrow, Chris Hegarty, Stephen Lowry and Jamie Richards made their debut for Linfield who had the ball in the net against B36 Torshavn on 11 minutes from Andy Waterworth but it was cancelled out by the linesman.
Linfield suffered a blow when centre half Mark Haughey had to retire injured, his place taken by Jimmy Callacher.
The visitors had started brightly and took the lead on 38 minutes when Aaron Burns rolled the ball into the path of Jamie Mulgrew and the midfielder fired the ball home from 25 yards out.
Michael Carvill had a big shout for a penalty on 57 minutes and then Andy Waterworth fired wide in the next attack.
The home team levelled with 18 minutes to go through Lawal but the night belonged to Linfield when Peter Thompson set up Michael Carvill and he added the finish from close range.
The second leg is at Mourneview Park on Tuesday, 8 July.
Stephen Cockcroft made an instant impact on his Crusaders debut scoring the opening goal on 23 minutes. Declan Caddell miscued his shot but Cockcroft was in the perfect place to fire home from close range.
The home side doubled their lead just before the break, goalkeeper Timofeyevas misjudged a cross from Craig McClean cross and Jordan Owens was able to head home.
Crusaders grabbed a third goal after the break on 57 minutes when Chris Morrow's shot was parried by the keeper and skipper Colin Coates popped up to head home the rebound.
Ekranas hit a late consolation through Susnjar near the end but the Crues are well on course to make the second round of a European tournament for the first time in their history.
Things were going according to plan for Glenavon in Iceland as they held FH Hafnarfjordur to 0-0 with just eight minutes to go.
However, fitness began to tell and the home side smashed three goals to make their journey to Lurgan next Thursday night a much easier task.
The Lurgan Blues played their part in an even first 45 minutes with Eoin Bradley and Kevin Braniff going close to breaking the deadlock.
In the second half the home side, unbeaten and midway through their season, began to look dangerous but it was Glenavon who almost took the lead with Kris Lindsay bringing a fine save out of the keeper.
FH broke the ice on 82 minutes and struck another two in added time to leave Glenavon with a mountain to climb in the second leg.
It was all smiles at the Brandywell as Derry City saw off Welsh side Aberystwyth Town 4-0.
Derry broke the deadlock on 14 minutes when Patrick McEleney beat his marker and drilled a low shot into the net.
City won a penalty when Danish referee, Mads-Kristofferson, sent off Aberystwyth's keeper for a challenge on Michael Duffy on 22 minutes and Rory Patterson made no mistake from the spot.
Mark Timlin scored a screamer from 30 yards to make it 3-0 and Barry McNamee sealed a great night with number four.
Thursday's final scores:
B36 Tórshavn 1-2 Linfield
B36 Tórshavn: Lawal 72.
Linfield: Mulgrew 38, Carvill 88.
Crusaders 3-1 Ekranas
Crusaders: Cockroft 23, Owens 43, Coates 57.
Ekranas: Šušnjar 88.
FH Hafnarfjördur 3-0 Glenavon
FH Hafnarfjördur: I. Óskarsson 82, Gudnason 90, 90+3.
Derry City 4-0 Aberystwyth
Derry City: P. McEleney 15, Patterson 25 (pen), Timlim 47, B. McNamee 86.