The Hoops travelled to the tiny Samsungvollur stadium with a 2-0 advantage from last week’s first leg in Glasgow but found themselves under some unexpected pressure after only six minutes when Olafur Karl Finsen shot the home side into a shock lead.
However, midfielder Nir Bitton eased the nerves when he levelled in the 32nd minute from a Stefan Johansen corner and when defender Charlie Mulgrew fired in a second early in the second-half it was all over as a contest with late goals from substitute Leigh Griffiths and Johansen making it 4-1 to see the Hoops through 6-1 on aggregate.
Ronny Deila’s side will play Qarabag in the third qualifier, with the first game at Celtic Park next week, and are sure to be tested to a greater degree against the Azerbaijan outfit who came through their tie against Montenegro’s Rudar Pljevlja.
There was no surprise that Virgil van Dijk returned to the heart of the Celtic defence after suspension but some eyebrows were raised when it was announced that Mulgrew would replace Emilio Izaguirre at left-back.
Nadir Ciftci, who was handed an eight-game domestic ban on Monday, with two suspended – for a biting incident on the final day of last season while still a Dundee United player – was again preferred in the main striking role to last season’s top scorer Griffiths.
It was the first Champions League qualifier held in Stjarnan’s 1,000-capacity ground with its one stand and plastic pitch.
In somewhat surreal surroundings, around 200 Hoops fans watched their side start positively enough but fall behind to a sloppily conceded goal.
The home side were allowed to get the ball forward far too easily and when striker Jeppe Hansen went past Van Dijk and picked out Finsen in the box, he slipped Bitton and drilled the ball past Craig Gordon.
Immediately the atmosphere changed and there was no hiding the trepidation among the small band of hitherto unconcerned Celtic supporters as they watched their side fight to get back in the driving seat.
In the 22nd minute Celtic defender Dedryck Boyata had the ball in the net following a Johansen shot which had been spilled by former Motherwell keeper Gunnar Nielsen, but the flag went up for offside.
However, the Glasgow side were not to be denied and just after the half-hour mark Bitton’s close-range header from Johansen’s delivery gave a flailing Neilsen no chance.
Midfielder Stuart Armstrong came close with an effort moments later and Bitton forced a save from Nielsen with a 25-yard drive as the Hoops side went looking to forge ahead.
The second goal came in the 49th minute when Mulgrew took a pass from captain Scott Brown before hammering a drive from 20 yards low past Neilsen and into the corner of the net.
The tie was effectively over with only the final score up for grabs.
Substitute Griffiths, on for Armstrong in the 62nd minute after defender Efe Ambrose had replaced Mikael Lustig, missed a decent chance when he knocked the ball past the far post, before testing Nielsen with a long-distance drive.
In the 68th minute Bitton prodded in a Mulgrew free-kick but was flagged offside.
Moments later Gordon made a fine save from Finsen’s powerful drive as the Icelanders tried to rally.
As the match stretched further, Nielsen did well to block an angled-drive from wide-man Gary Mackay-Steven, who set up Griffiths in the 88th minute for the former Wolves striker to sweep the ball into the net, before Johansen converted a Mulgrew cut-back in added time to put an even better gloss on the result.