The hosts created the first opening of the match inside 10 minutes when Stefan Johansen pounced on a loose pass and found Nadir Ciftci, but the debutant striker was denied by Gunnar Nielsen in the Stjarnan goal.
As expected, it was Celtic who were asking the majority of the questions, and they squandered another clear chance to break the deadlock midway through the half when Johansen dragged a shot wide after the visitors had failed to clear Charlie Mulgrew's cross.
The pressure continued to build on the visitors, but they held firm despite Stuart Armstrong sending a long-range effort fizzing just over before drawing a fine stop from Nielsen.
Ciftci then came close again with a glancing header as Celtic enjoyed total domination but struggled to break through their resilient opponents.
However, the visitors were unable to hold on until half time as Celtic finally opened the scoring a minute before the break when Dedryck Boyata powered a header home on his debut.
The home side were quick to reassert their authority in the second half when Ciftci nodded narrowly over, although they were given a scare at the other end by a Jeppe Hansen effort that went narrowly wide.
Ciftci was then denied again by Nielsen, but Celtic got their second goal 10 minutes into the half when Johansen scooped the ball beyond the keeper having latched on to a through-ball.
The Bhoys pushed hard for a third goal and they were handed a glorious chance to extend their lead when Johansen was fouled inside the box, but substitute Leigh Griffiths saw his well-struck penalty kept out by Nielsen.
Ronny Deila's side almost made it three in the dying stages when Johansen thumped an effort goalwards having been found by Armstrong, only to be denied by a fine stop from Nielsen.
They were forced to settle for just the two goals, however, and will now head to Iceland for the second leg next week looking to close out the victory.