At one stage, Astana were within three goals of completing one of the greatest comebacks of all time, while Stanimir Stoilov's side missed numerous chances to heap further pressure on Celtic.
However, Brendan Rodgers's team netted twice in the final 10 minutes to eventually run out 8-4 victors on aggregate and avoid similar European embarrassment to that of Glasgow rivals Rangers.
With a five-goal advantage in their favour, there would have been concerns that complacency could set in for the trip to Kazakhstan, but Celtic began well with James Forrest having an early header saved by the goalkeeper.
Astana were always a danger going forward with Patrick Twumasi providing a frequent threat for the home side, but Brendan Rodgers would have been satisfied with how his players dealt with the first quarter of the contest.
Celtic were back on level terms seven minutes later when Scott Sinclair curled a delightful effort into the corner, but the Scottish outfit almost conceded a second before half time, with Gordon having to be alert to keep out Marin Anicic's header.
Ahead of the restart, Astana were aware that they required a minimum of six goals to keep their hopes alive, and within five minutes of the beginning of the second half, a third of the Blues and Yellows' target had been achieved.
Celtic were wilting by the minute at the back and only some wayward finishing ensured that Astana did not net a third goal in quick succession, one which would have left Rodgers's side increasingly nervous in their defensive third.
Rodgers made three substitutions in an attempt to settle the match down but with 21 minutes left, Astana and Twumasi scored another as the Ghanaian rifled a low shot under Gordon from an acute angle.
However, with 10 minutes left, Celtic all but guaranteed their progress as Olivier Ntcham received the ball on the left-hand side of the box and sent a deflected effort inside the near post.
It was relief for Rodgers and the club's travelling fans and although they failed to get back on level terms on the night, Griffiths added a late third with a well-struck effort into the corner from 14 yards.