Ronny Deila's men improved after the break but aside from a great save by Qarabag keeper Ibrahim Sehic from a Virgil van Dijk header, they posed little threat until the 82nd minute when the former Manchester City defender, recruited in the summer, headed in a corner from substitute Kris Commons.
The Scottish champions have the goal to defend in the return game in Baku next Wednesday night - and they will be grateful for it - although on the evidence of the first-leg the tie is far from over.
Deila's men had beaten Icelandic part-times Stjarnan 6-1 on aggregate in their previous qualifier but Qarabag delivered on their promise to be a much tougher obstacle.
Hoops fans, though, were boosted beforehand by the news that keeper Craig Gordon had extended his contract until 2018.
There was no surprise that Emilio Izaguirre returned to left-back to replace the injured Charlie Mulgrew.
The main talking point was Deila again preferring summer signing from Dundee United, Nadir Ciftci, to last season's top scorer Leigh Griffiths, who started on the bench, along with Commons, back after a toe injury.
It was not one of the big European nights at Celtic Park in terms of atmosphere but the home supporters tried their best to increase the tempo amid a flat start to the match.
Qarabag, neat and tidy at times in possession, were far from under siege and they threatened on the break.
Celtic's first real attempt on goal came in the 24th minute when midfielder Gary Mackay-Steven's drive from the edge of the box was grabbed by Sehic at the second attempt.
Five minutes later, in another Qarabag break, wide-man Javid Taghiyev's pass to Richard Almeida at the edge of the penalty area was swept just past Gordon's right-hand post by the Brazilian midfielder.
Green and white frustrations grew as the match edged towards the interval.
Celtic might have been punished had Alharbi El Jadeyaoui not over-hit his cross from the right with the Hoops defence scrambling to get back, before Gordon made an unorthodox save from Almeida's whipped-in free-kick, after Van Dijk had been booked for a foul on Taghiyev.
Celtic's start to the second-half revealed more zest if not more potency.
Ciftci headed an Izaguirre cross wide of the target before smashing the ball straight at Sehic from 12 yards after getting a break on the ball.
Winger James Forrest replaced Stuart Armstrong just after the hour mark before midfielder Nir Bitton made for way Griffiths.
In between, Sehic made a fantastic save from a powerful Van Dijk header, from a Stefan Johansen free-kick, before the Dutchman arrowed a long-range drive just over the bar.
Commons came on for Ciftci with 10 minutes remaining and, when he whipped in a corner from the right, Boyata, who scored in the previous round at home to Stjarnan, rose high to bullet his header past Sehic, with the Qarabag keeper claiming he had been fouled to no avail.
Celtic will unfurl the Scottish Premiership flag against Ross County on Saturday as the league season begins but they still have a bit of work to do next week to remain in the Champions League.
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