The Scottish champions dominated the possession, but did not find the back of the net until the 84th minute.
The second leg will take place at Murrayfield next week.
Read how all the action unfolded below:
Good evening! It's just under half an hour until Celtic begin their Champions League campaign against Icelandic champions Kr Reykjavik. The two clubs have already announced their teams, so let's start with a look at the lineups...
REYKJAVIK STARTING XI: Magnusson, Hauksson, Gunarsson, Saevarsson, Josepsson, Siguardarson, Atlason, Sigurdsson, Martin, Finnbogason, Balbi
CELTIC STARTING XI: Forster, Lustig, Izaguirre, Van Dijk, Ambrose, Griffiths, Mulgrew, Stokes, Commons, McGregor
REYKJAVIK SUBSTITUTES: Jonsson, Gunarsson, Ragnarsson, Jensson, Ormarsson, Sigurjonsson, Arouna
CELTIC SUBSTITUTES: Matthews, Boerrigter, Pukki, Zauska, Kayal, O'Connell, Henderson
The Scottish champions are without club captain Scott Brown, who injured his hamstring during a pre-season match earlier this month, while James Forrest also misses out with a groin injury. However, Fraser Forster, Emilio Izaguirre and Efe Ambrose are all included, despite not having featured during pre-season for the Hoops as a result of their involvement in the World Cup.
The majority of Reykjavik's squad are Icelandic players, but English striker Gary Martin, who came through Middlesbrough's youth system, starts for the home side this evening. He's scored 30 goals in 67 appearances for the club since joining in 2012.
Ahead of this match, Celtic assistant manager John Collins told reporters: "We have lots of other good players in the squad who have to step up and show that they can do a job for the team as well and I have no doubt they will. That's three weeks that we have been back now, this is the big one. We want to go there and try to score goals. We have watched them twice, we have had videos cut up so we know their strengths and weaknesses but at the end of the day it is about how we play."
Meanwhile, KR Reykjavik manager Runar Kristinsson said: "We are all amateurs here and they are fully professional so there's a lot of difference, I don't think we have a lot of chance but it's football, it's 11 v 11. We can go out and try to make it as hard for them as we can to try and hopefully get some result that we can take with us to Edinburgh. The players love these games, they love to play in Europe, it gives the team a boost."
Kristonsson's reference to Edinburgh isn't a mistake, Celtic will have to play the second leg of this tie at Murrayfield due to their stadium being used as part of the Glasgow Commonwealth Games.
While Celtic have only just begun their pre-season preparations, the Icelandic league, the Urvalsdeild, has been underway for a couple of months. KR, who are the current champions, currently sit third in the table, having won six of their 10 matches so far.
PREDICTION! With just a couple of minutes until kickoff, it's time to make a prediction. On paper, this should be a comfortable evening for the Scottish champions, but KR have been in good form domestically. I'm going for a 2-0 win to Celtic.
The players are just starting to make their way out ready for kickoff. The KR Vollur only holds 2,781, but it's pretty much at capacity this evening.
KICKOFF! KR get the match underway.
Celtic try to put the hosts under pressure straight away as Leigh Griffiths collects the ball on the edge of the penalty area, but he can't control the ball and Magnusson is able to gather.
A clumsy challenge on McGregor from Josepsson gives the visitors a free kick in a dangerous area, which Mulgrew curls in towards Commons, but KR are alert to the threat and manage to make the clearance.
A quick burst from Stokes allows Commons to cross in towards Griffiths, but Gunnarsson gets to the ball first and heads clear. It's been all Celtic so far, but they're yet to really test goalkeeper Stefan Magnusson.
OFF THE BAR! What a wonderful strike from Kris Commons! The striker hits the ball first time with a 25-yard volley after being set up by Griffiths, but his efforts is just kept out by the crossbar.
CHANCE! Celtic are really starting to come to life now as Johansen finds Commons who has another decent effort from range, but Magnussen just turns it wide for the hosts to give Celtic their first corner of the match.
CHANCE! From the corner, Ambrose tries a shot from the centre of the box, but Josepsson is back on the line to keep it out.
Reykjavik are putting Celtic under pressure in their box for the first time this evening as Gunarsson makes a superb run down the right flank before crossing in looking for Martin, but Forster eventually manages to claim the ball.
Atlason manages to win the ball for the hosts in midfield and quickly slots it through to Martin, but the English striker's shot is cleared by Lustig. He has another chance a couple of moments later, but is flagged for offside. The Icelandic side have looked a little more threatening in the last couple of minutes.
KR win a free kick in midfield when Johansen brings down Martin, but Balbi's ball towards Martin is dealt with by Lustig. Celtic immediately look to press down the other end, but McGregor's effort into is comfortable taken by Magnusson.
CHANCE! Sigurdsson finds Martin with a well-placed through ball and the striker is given far too much space by Ambrose, allowing him to drill the ball towards goal, but Forster gets down low to make the stop for the visitors.
A clumsy challenge from Siguardsson gives the visitors a free kick, which Mulgrew collects on the edge of the box, but Siguardsson makes amends for his earlier error by quickly clearing. Celtic are continuing to see the majority of the ball, but they're finding it hard to get the better of a well-organised back four.
Reykjavik are on the attack again as Martin crosses in from the right looking for Gunnarson, before Sigurdsson threads it back in, but Lustig clears the initial danger before Izaguirre boots it away to set Griffiths up for a chance at the other end.
Baldi is brought down around 35 yards out, and it's Gary Martin lining up the free kick, but the striker's curling effort lacks power, and is comfortably dealt with by the wall.
Celtic patiently play their way through midfielder before Izaguirre launches the ball into the box, but it's cleared. Mulgrew wins it back in midfield to set up Stokes, but his shot goes just wide to give the hosts a goal kick.
The hosts win a free kick as Johansen drags down Balbi in the centre circle. Sigurdsson launches it deep looking for Martin, but Celtic quickly get on the counter looking to give Commons a chance from the edge of the box, but Reykjavic continue to be tight at the back and make sure that the threat is cleared.
Commons and Griffiths try to put Reykjavic under pressure during a throw in, but Atlason manages to get the ball to Martin. The English forward has certainly been the home side's best chance of a goal in this first half.
CHANCE! Johansen delicately chips the ball over for Lustig, who tries to flick it on the Commons, but Josepsson manages to put it behind for a corner. Mulgrew delivers the set piece, but his cross goes straight out the other side to Hauksson.
HALF TIME: KR Reykjavic 0-0 Celtic
Kris Commons tries to apply some late pressure, but he trips on the edge of the box and the referee blows his whistle to end the half with no stoppage time.
Kris Commons came closest to opening the scoring when he struck the crossbar early on, but Gary Martin has looked threatening up front for the hosts. While Celtic have certainly had the majority of the chances, Reykjavic have not allowed them to control the game.
Let's take a look at some of the statistics from the first half. Celtic dominated possession with 74%, and had six shots to Reykjavik's two, although both sides had one on target. Celtic have had three corners, while the hosts are yet to have one.
Ronny Deila has sent the Celtic side back out early for the second half, they'e already been out for a couple of minutes, while the hosts are just making their way back now.
KICKOFF! The second half is underway
There were no changes for either side at half time. Reykjavik have made a solid start, with Atlason putting the Celtic back four under pressure, forcing Lustig to concede the first corner for Celtic. However, the Van Dijk is fouled as the ball is delivered to let Celtic off.
CHANCE! Griffiths quickly bursts forward before finding Commons, who unleashes a powerful strike from just inside the box, but it just creeps wide.
Celtic have a corner of their own, which Stokes delivers to the head of Lustig. His effort is cleared but if falls kindly for Griffiths, but he mis-hits his shot and fails to trouble Magnusson in the home goal.
CHANCE! Reykjavik fails to clear from a Stokes cross, allowing Commons another shot, but once again his effort is a fraction off-target. The striker has been lively all night, but hasn't really but Magnusson under any pressure.
CHANCE! Commons does manage to force a save from Magnusson, but his shot was scuffed, giving the keeper a comfortable hold. Celtic certainly seem to be getting closer though.
Celtic win a free kick in a dangerous area, which Stokes elects to take, but his curling effort is a fraction high.
SUB! The hosts are the first to make a change, bringing on midfielder Farid Zato for Gonzalo Balbi Lorenzo, who incidentally is the brother in law of Luis Suarez!
CHANCE! How on earth did that not go in?! Leigh Griffiths hits a sensational strike from out wide, which rattles off the underside of the crossbar onto the goalline before bounding away from the goal, allowing Hauksson to put it out for a corner.
Mulgrew curls the corner is for Celtic, but Magnusson comes off his line to gather the ball. Celtic have really upped their intensity since the start of the second half, but just can't find a way of opening the scoring.
Celtic are on the move again as Griffiths carves a path through the Reykjavik defence, but once again the finishing touch is missing as he scuffs the shot well wide.
BOOKING! Emil Atlason is the first player to be shown a yellow card. It didn't appear to be for anything that happened on the ball so he must have said something that the referee wasn't happy with.
Griffiths is on the move again, but Sigardarson does enough to stop him getting his shot away, but does concede the corner. Mulgrew delivers it into the box, and Stokes jumps for the ball, but can't quite get to it.
Reykjavik are on the counter attack with Martin trying to get his shot away, but Virgil van Dijk makes a solid challenge to put the ball behind for a corner, and Forster manages to claim the ball from the resulting set piece.
SUBS! It's a double change for Celtic as Leigh Griffiths is replaced by Derk Boerrigter, while Anthony Stokes makes way for Teemu Pukki, who has been in strong goalscoring form in pre-season.
OFF THE POST! What a huge let-off for Reykjavik as Commons forces a save from Magnusson, but the keeper parries the ball into the path of Pukki. The substitute has an open goal, but has barely had a touch since coming in, and somehow fires his first shot of the night off the post and behind for a goal kick.
Celtic are still pressing but Kris Commons is startling to look tired and launches a long-range effort well wide of the target.
SUB! Thorsteinn Ragnarsson comes on for the final 10 minutes in place of Gary Martin, who had some good chances in the first half for the hosts, but has been kept pretty quiet in the second half.
CHANCE! Zato, who has shown impressive pace since coming off the bench, sets up a chance for the home side, but Finnbogason and Ragnarsson both jump for it and seem to put each other off, allowing Forster to claim it.
CHANCE! Celtic quickly win a corner down the other end which Commons takes short before crossing in for Pukki, but the sub doesn't quite catch the half-volley right, allowing Magnusson to parry before gathering it at the second time of asking.
GOAL! KR Reykjavik 0-1 Celtic (McGregor)
Callum McGregor has found the back of the net on his competitive debut for Celtic, superbly rounding a couple of defenders out wide on the edge of the box before drilling a low shot past Magnusson. What a fantastic strike from the 21-year-old.
The home side try to get one back straight away as they burst into the Celtic box. Zato has the ball on the left flank and looks to cross, but there's no-one in the box to help him out, allowing Lustig to clear.
Celtic want a second as McGregor wins the ball again, before Commons launches the ball forward for Lustig, but the defender had crept off side during his run into the box.
SUB! KR make their final change with Almar Ormarsson replacing Emil Atlason.
BOOKING! Jonas Saevarsson picks up a yellow card for a reckless challenge.
There match is being played at a frantic pace right now as both sides desperately seek a late goal. With the away goals rule applied in this competition, a second for Celtic would leave the Icelandic champions needing three next week, while an equaliser for KR would keep them in the tie. Both sides have three minutes of stoppage time to find the back of the net.
Lustig makes a stunning run forward to set up Commons, but the striker's effort is blocked, and that should be about all we have time for.
FULL TIME: KR Reykjavic 0-1 Celtic
Callum McGregor's late strike gives Ronny Deila a win in his first competitive match in charge of Celtic, and gives the Scottish champions an away goal in the first leg of their second-round Champions League qualifier against KR Reykjavic.
It arguably should have been more for the visitors, who were dominant throughout and had a mighty 27 shots compared to their opponent's three.
The sides will meet again next week at Murrayfield, with Celtic's stadium being used for the opening ceremony at the Commonwealth Games. Having played in front of a crowd of just over 1,500 tonight, the 67,000 capacity stadium will provide a completely different challenge for the Icelandic champions as they seek to get back into this tie.
Well that's all from this match, but stick around to read our match report, as well as reaction to the game. Thanks for joining me this evening, goodbye.