Patrick Twumasi scored twice as Astana, beaten 5-0 last week in Glasgow, threatened an incredible comeback before Brendan Rodgers' side secured their place with an 8-4 aggregate victory.
Kristoffer Ajer diverted a Dmitri Shomko shot beyond Craig Gordon as Astana took the lead on 26 minutes, but Scott Sinclair settled Celtic nerves with a superb equaliser.
However, Serikzhan Muzhikov drilled Astana back ahead on the night before Ghanaian striker Twumasi added a brace to leave Celtic rattled and facing a tense final 20 minutes.
Olivier Ntcham ended Astana's hopes though when his deflected strike made it 4-2 before Leigh Griffiths reduced the deficit further in the final minute.