A late goal from Callum McGregor, on his debut, was enough to make the second qualiying round second leg in Scotland next week little more than a formality.
New manager Ronny Deila's first competitive game in charge turned out to be a stroll in the unprepossessing setting of the KR-vollur stadium.
The Norwegian, who only took over from Neil Lennon last month, saw his side control the first half, with Kris Commons hitting the bar, but they could not convert their easy superioirty into goals.
Leigh Griffiths and substitute Teemu Pukki also hit the woodwork in the second half before 21-year-old McGregor, making his first competitive start for the Hoops after impressing in pre-season, drilled a shot low past home goalkeeper Stefan Magnusson with seven minutes remaining.
But their lack of firepower will have set the alarm bells ringing for Deila and he will know that if the Hoops are to make the main Champions League draw, they will have to sharpen up their act in front of goal.
Commons, top scorer last season with 32 goals, rattled the bar from 25 yards with a dipping volley and moments later, forced a good save from Magnusson with a powerful drive.
In the 74th minute Deila introduced Pukki and Derk Boerrigter for Stokes and Griffiths in his bid to find a goal.
As the home side found themselves with their backs against the wall, Pukki hit the post after Magnusson had made a double save from Commons and McGregor.
The visitors kept pushing for the opener and Boerrigter was the next Hoops player to be foiled by Magnusson before McGregor turned past three players to fire in a low drive.
With Celtic Park out of commission due to the 2014 Commonwealth Games, the return fixture will be at Murrayfield in Edinburgh next Tuesday night.