It means the Scottish champions will take a commanding advantage to Kazakhstan for the return leg next Tuesday.
An own goal from Evgeni Postnikov and two goals from Scott Sinclair either side of half-time had the Hoops cruising to victory before James Forrest's strike and a late Igor Shitov own goal rounded off the rout.
It means the Scottish champions will take a commanding advantage to Kazakhstan for the return leg next Tuesday and puts them within touching distance of sealing their place in the lucrative group stages for a second successive season.
"We've had a number of outstanding performances since I came in but this was a different game and a different occasion," Rodgers said.
"Every game in this qualification is a pressure game. The pleasing aspect is how the players coped with that - their calmness in possession, how we circulated the ball and how we opened up the spaces and tired Astana out."
"It had everything in it -- creativity, pressing, a real good structure to the team and good organisation."
"Some of our play was outstanding."
"We scored five goals and missed one or two with that. For a Scottish side to do that at this level of competition against a side that a couple of years ago were drawing with Atletico Madrid and Benfica and show that level of composure makes me really proud of the team."
"It was a brilliant performance. Very complete in many aspects of how we work."
It was a marked improvement from last season when the Hoops narrowly saw off the Kazakhstan champions 3-2 over two legs in their third round qualification match of the same competition.
And Rodgers, who led his side to an unbeaten domestic season as they collected a treble, praised the progress his side have made since his arrival at Celtic Park last summer.
"It was a great demonstration of the work that the players have made and how they have progressed over the course of the last season," the Celtic manager said.
"We made a big step forward last year. We had a physical courage to work but over the course of the season we played a game that you have to be able to play under pressure."
"That progress is continual and I think you see tonight the marked difference from when we won here last season 2-1."
"There is a maturity in performance, an excitement and creation of chances."
"Tactically the players are thinking about football. It's not just about going out and running about and hoping."
"We are seeing that development and it is great to see young Scottish players in particular having that composure, quality, hunger and desire to play the game to that level."