Match ends, Celtic 2, Stjarnan 0.
Celtic are well placed to progress in their Champions League qualifier after a night of domination and frustration against Stjarnan in Glasgow.
Three good chances were spurned before debutant defender Dedryck Boyata headed home from corner on 44 minutes.
Stefan Johansen then tucked home Stuart Armstrong's clever cut-back.
But there was a sour note when substitute Leigh Griffiths had a penalty kick saved after squabbling with team-mate Johansen for the ball.
Former Motherwell goalkeeper Gunnar Neilsen did well to push the effort away and made a series of good saves to keep the Icelanders in the tie ahead of next week's return leg.
As expected, Celtic started on the front foot, knocking the ball around with a swagger that suggested they were comfortable with the favourites' tag.
New signing Nadir Ciftci was chosen to lead the home attack and probably should have scored after nine minutes.
With back to goal, the Turk cushioned an Armstrong pass just inside the box but his tame effort on the turn was stopped by Neilsen.
Ronny Deila's side continued to probe with James Forrest and Armstrong delivering from the wings but Stjarnan were defending in numbers and high balls were mopped up with relative ease.
Stjarnan's intentions were clear as they settled into a formation that was designed to frustrate and there were grumbles from the crowd as Johansen blasted just wide from eight yards.
The groans got louder after Armstrong danced through the visiting defence only to see his effort hit the outstretched foot of Neilsen.
Painful memories of last year's qualifying defeats to Legia Warsaw and NK Maribor may have been fuelling fans' desire to kill off the side who ended Motherwell's Europa League dreams last season.
For all their hard work there was little doubt that the Icelanders were also enjoying some good fortune.
But that luck ran out just as the clock clicked towards half-time.
Johansen's corner sailed over the outstretched fingers of Nielsen and onto the grateful head of Boyata, who was lurking at the back post to guide the ball into the back of the net.
The relief inside Celtic Park was palpable and, had Forrest shown more composure in front of goal a minute later, the pressure would have been off completely.
Having found a way through Stjarnan's wall of blue, the winger pulled the trigger from 10 yards but his effort was deflected wide.
The breakthrough did little to alter the visitors' game plan but they turned defence into attack just moments after the break and came very close to a shock equaliser when Jappe Hansen's shot trickled wide.
Hearts remained in mouths as the visitors enjoyed a good five-minute spell, with two efforts turned over the crossbar.
But when the home fans dared to peer between cracked fingers, Armstrong delivered the second goal their frayed nerves were craving.
Johansen started the move and ran on to the former Dundee United midfielder's pass to deftly slot the ball beyond Neilsen.
The Celtic onslaught intensified as Armstrong and Johansen continued to link up well but a great opportunity to add a third goal was squandered after scenes Deila described as "unprofessional".
Johansen was upended to win a penalty and wanted to take the spot-kick before relenting to Griffiths's demands.
Nielsen saved low to his right and produced a better stop in stoppage time to repel Johansen's close-range shot.
- 20BoyataSubstituted forAmbroseat 82'minutesBooked at 88mins
- 49ForrestSubstituted forGriffithsat 58'minutes
- 7CiftciSubstituted forStokesat 74'minutes
- 1NielsenBooked at 62mins
- 7JohannssonSubstituted forRunarssonat 74'minutes
- 11BjörgvinssonSubstituted forPalssonat 86'minutes
- 23BjörnssonSubstituted forGunnarssonat 81'minutes
- 22Kari Knutsson
- Daniel Siebert
- Shots on Target
Second Half ends, Celtic 2, Stjarnan 0.
Attempt missed. Anthony Stokes (Celtic) right footed shot from outside the box misses to the right.
Efe Ambrose (Celtic) wins a free kick in the attacking half.
Foul by Jeppe Hansen (Stjarnan).
Corner, Celtic. Conceded by Gunnar Nielsen.
Attempt saved. Stefan Johansen (Celtic) left footed shot from the centre of the box is saved in the top right corner.
Efe Ambrose (Celtic) is shown the yellow card.
Foul by Efe Ambrose (Celtic).
Jeppe Hansen (Stjarnan) wins a free kick in the defensive half.
Substitution, Stjarnan. Atli Palsson replaces Arnar Mar Björgvinsson.
Foul by Efe Ambrose (Celtic).
Veigar Gunnarsson (Stjarnan) wins a free kick on the right wing.
Mikael Lustig (Celtic) wins a free kick on the left wing.
Foul by Ólafur Karl Finsen (Stjarnan).
Substitution, Celtic. Efe Ambrose replaces Dedryck Boyata because of an injury.
Delay over. They are ready to continue.
Delay in match Dedryck Boyata (Celtic) because of an injury.
Substitution, Stjarnan. Veigar Gunnarsson replaces Halldor Orri Björnsson.
Corner, Celtic. Conceded by Heidar Aegisson.