A scrappy strike from the edge of the area which took a deflection off defender Denis Dedechko after 29 minutes will boost the confidence of Martinez who has struggled to live up to his 35 million-euro ($39.7 million) transfer fee from Porto.
Saul Niguez opened the scoring with a flick from the edge of the six-yard box after 22 minutes and Oliver Torres delicately chipped the ball over keeper Nenad Eric after 63 minutes.
A Juanfran Torres cross was turned into his own goal by Dedechko to complete the rout a minute from time.
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Atletico Madrid have six points from three games in the group while Kazakhstan's Astana have one.
Diego Simeone’s side had plenty of possession but their passing through the middle of the pitch was not precise early on and most danger came from set-plays.
Astana coach Stanimir Stoilov left eight players behind in Kazakhstan with their priority to keep them fresh for their bid to win their domestic league.
Atletico's Diego Godin went close with several headers from corners and it was no surprise that the breakthrough came from another corner, played short to Yannick Ferreira-Carrasco and his cross was clipped in at the near post by Saul.
The Spaniards began to probe the Astana area before Martinez got the second and they continued to pressure with Martinez again going close with a header and Carrasco had a shot well-saved.
Torres scored with a sweet finish having been set up by Gabi Fernandez and Dedechko completed an unfortunate night with an own goal.
(Reporting by Tim Hanlon in Barcelona, editing by Ed Osmond)
