The Norwegian's first competitive game since taking over from Neil Lennon last month took place in the somewhat surreal surroundings of the KR-vollur stadium in Iceland, which housed only around 1500 fans in its one stand, approximately 200 of whom were away supporters.
The Scottish champions controlled the first half, with attacking midfielder Kris Commons hitting the bar, but they could not fashion the breakthrough.
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.
With Celtic Park out of commission due to the 2014 Commonwealth Games, the return fixture will be at Murrayfield in Edinburgh next Tuesday night. And while Celtic will surely go through to the next qualifier, Deila will know improvements are required, particularly in front of goal, if they are to achieve their goal of reaching the group stages.
It was an awkward night for Celtic.
The home side had former Parkhead youth player Kjartan Finnbogason backing English striker Gary Martin, but they struggled to cope with the movement of the Parkhead men and were let off the hook in the ninth minute when Commons, top scorer last season with 32 goals, rattled the bar from 25 yards with a dipping volley.
Moments later, the former Nottingham Forest and Derby player stung the palms of Magnusson with a powerful drive from further out but the Glasgow side were unable to capitalise on the corner.
As the first half unfolded, Celtic remained on top with only occasional threats from the Icelandic champions, one of which almost bore fruit in the 23rd minute when Martin sent in a shot from 25 yards which drew a good save from goalkeeper Fraser Forster, starting his first game since returning from World Cup duty with England.
However, there was no doubt that Runar Kristinsson's team were slowly growing in confidence as Celtic's early enthusiasm dissipated.
In the 32nd minute striker Anthony Stokes worked a one-two with Commons at the edge of the box but ballooned his volley high and wide, summing up their first-half dip.
Commons, the most likely of Deila's attackers, came close with two attempts early in the second half, both times his shots clearing the crossbar.
The Celtic player then had a shot from 10 yards saved by Magnusson after being set up by Stokes as Celtic looked to be wearing down the Icelandic side.
It seemed a matter of time before the Hoops would get in front.
Stokes fired a 30-yard free-kick just over before Griffiths smacked the underside of the bar with a vicious drive from the edge of the box which had the KR Reykjavik keeper clawing at air.
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 Celtic player to be foiled by Magnusson before McGregor turned past three players to fire in a low drive which found the net to the relief of the visiting fans and, no doubt, Deila.