The visitors trailed Astana for most of the first leg encounter after Yuri Logvinenko headed his side into an 18-minute lead.
Astana had chances to build on their advantage and would later come to regret those spurned opportunities when Leigh Griffiths fired into the bottom corner following great work by Patrick Roberts.
The away goal will give Celtic an advantage when the two teams meet again in Glasgow next Wednesday.
Brendan Rodgers side started the brighter of the two teams as Griffiths went close with a shot from the edge of the area.
However, they would fall behind to Astana’s first threatening attack. Efe Ambrose didn’t properly clear a Patrick Twumasi cross and when the Ghanian curled in the resulting corner kick, Logvinenko took advantage of poor marking by Eoghan O’Connell and a mistake by Craig Gordon to head his team in front.
From there Astana took control of the game and could have added to their lead before the half.
Azat Nurgaliev forced Gordon into a terrific save after latching on to a ball into the penalty area, while Junior Kabananga also brought in the best in the Celtic stopper after getting on the end of a Twumasi header.
Twumasi went close himself minutes into the second period after firing a sloppy Stuart Armstrong pass towards goal. Gordon was troubled by the movement on the ball but managed to keep it out. Nuragliev then came the closest of the lot when his effort from the edge of the area rattled the top of the crossbar.
Rodgers changed up his team in order to stem the Astana flow, going to a 4-5-1 with Nir Bitton replacing the quiet Moussa Dembele.
The change had the desired effect of halting the hosts momentum, though an equaliser still seemed unlikely prior to the 78th minute when Roberts nicked the ball off Dmitriy Shomko at the byline. The youngster continued down the line before looking up and picking out Griffiths, and Celtic’s leading goalscorer last season made no mistake from 15 yards.
Astana had two great chances to put themselves back in front. Kabananga forced Gordon into a front post stop after springing the offside trap. Then, in injury time, Roger Canas wasted the chance of the match by heading over from four yards after drifting into an unmarked area.