An own goal from Kristoffer Ajer opened the scoring for Astana after Celtic had missed a couple of early opportunities.
The tie looked to be over as a contest when Scott Sinclair netted a superb goal for the visitors, curling into the top corner from just inside the area.
However, Astana came roaring out at the start of the second half, with Patrick Twumasi and Serikzhan Muzhikov both netting for the hosts within four minutes of the restart.
Twumasi then really cranked up the nerves with a goal on the 69th minute. The Ghanaian then went close again a short time later.
The game was finally killed as a contest when Olivier Ntcham found the back of the net after a successful counter, before Leigh Griffiths completed the scoring in the 90th minute.