Rovers take their deserved lead, and with that vital away goal, into next Thursday’s second leg at Tallaght Stadium as Czech side Mlada Boleslav await the winners in the second round.
Though Stjarnan started better, with skipper Baldur Sigurdsson powering a header over before Johann Laxdal wastefully blazed off target, Rovers passed their way into the game, troubling the home defence for the first time on 15 minutes.
Simon Madden’s ball in from the right was loosely headed away by a defender. And though Brandon Miele latched onto it, Daniel Laxdal bravely blocked the shot from just inside the area.
Shaw then looped a header over the crossbar minutes later from Brandon Miele’s volleyed cross as Rovers visibly grew in confidence.
Stjarnan’s Haraldur Bjornsson was the first keeper to be worked just past the half hour.
Trevor Clarke’s tenacity set up Ronan Finn whose drive from distance was tipped over by the home keeper. Seconds later, Madden brought the best out of Bjornsson again with a stinging shot the keeper turned round a post.
The pressure then told on 38 minutes as Rovers got their deserved lead.
Miele’s corner from the right was only flapped away by Bjornsson. Dave Webster’s effort was blocked before the ball broke for Shaw to stab home from close range.
Rovers were well on top now and Finn brought another diving save from the overworked Bjornsson just before the break.
The second half was less eventful, the first opening not coming until 20 minutes in as Strjarnan tried to chase the game.
Fridjonsson found space to unleash at drive that Tomer Chencinski pushed away for a corner, even if it looked going wide anyway.
Chencinski then did make his first telling save of the game on 70 minutes, tipping away one-handed from Josefsson’s shot following a bit of a scramble in the Rovers’ area.
Bjornsson; J Laxdal (Aegisson, 87), D Laxdal, Gudjonsson, Josefsson; Hauksson, Sigurdsson, Hedinsson; Fridjonsson (Finsen, 81), Baldvinsson, Halldorsson.
Chensinski; Madden, Lopes, Webster, Byrne; Miele, Finn, Connolly (Bone, 74), Clarke; Burke (McAllister, 81); Shaw (O’Connor, 87).
Tomasz Musial (Poland).