The Parkhead men led from the first leg of the third qualifier through a Dedryck Boyata goal, but did not arrive at the Tofiq Bahramov Stadium with only defence on their mind.
In an open first half, the visitors gave as good as they got and amid a dearth of clear-cut chances at either end, Hoops keeper Craig Gordon make a fine save from Brazilian midfielder Richard Almeida.
The home side enjoyed plenty of possession again after the break but most of it in front of a resolute Hoops defence with the Scotland keeper seeing little action as Ronny Deila's side worked all the way to keep their lead intact and indeed, they might have snatched a goal in the closing minutes.
The draw for the play-off takes place on Friday and Norwegian boss Deila, in his second season in charge, might just feel he can get his side into the lucrative group stage this year.
An enthusiastic home crowd of close to 30,000, amid a heavy security presence, waved white Qarabag flags and drowned out the small band of Hoops supporters as the Azerbaijan champions looked to strike the first blow.
Despite the 9.30pm kick-off local time, there was enough heat and humidity to test the visitors, and the poor surface, scorched by high temperatures, made it difficult for both teams to get the ball down and play.
Qarabag bossed the early stages but the Parkhead defence, marshalled by reported Southampton target Virgil van Dijk, looked compact and protective of keeper Gordon.
Gradually the Scottish champions began to edge forward and in fact, play sped from end to end, helped by an often lenient Swedish referee, Martin Strombergsson.
After earlier causing a moment of panic with a poor punch, Gordon more than made up for that when he flew to his right to parry a powerful 25-yard drive from Almeida and that was as good as it got for Qarabag.
A water break after half an hour gave both sides a chance to regroup before Van Dijk made a clearing tackle in the six-yard box to thwart Dani Quintana.
An optimistic long-range attempt by Nir Bitton just before the break was saved by home keeper Ibrahim Sehic and there was enough for Deila to be pleased about at the interval.
There were some anxious moments, though, for the Glasgow side at the start of the second half.
Gordon was more than equal to the lobbed attempt by Alharbi El Jadeyaoui from outside the box.
But a sense of diminishing goal threat from Qarabag grew, as did the tension as the match entered the last 30 minutes.
In the 67th minute Leigh Griffiths replaced Nadir Ciftci, who had worked hard enough without really threatening, before Gary Mackay-Steven made way for Kris Commons
In the final stages, and after the second water break, Qarabag kept pressing.
But in the 83rd minute Celtic midfielder Stefan Johansen should have scored, instead firing just past the post from 12 yards after being set up by Stuart Armstrong.
Substitute James Forrest, on for Armstrong, then failed to pick out Griffiths on a Celtic break when the right pass would surely have led to a goal.
Ultimately, that late profligacy meant little as the Glasgow side held out for a nervy five minutes of added time before celebrating a good night's work.
Source : PA
Source: PA