First-half goals by the diminutive forward and 19-year-old debutant Kyriakos Papadopoulos put Fernando Santos's team firmly in charge at the Georgios Karaiskakis stadium.
Michael Mifsud pulled one back for Malta just after half-time but Fetfatzidis calmed the home side's nerves with his second of the night just after the hour to seal the three points.
Malta struggled to get hold of the ball as Greece continued to press forward, and the home side were 2-0 up in the 26th minute when Kyriakos Papadopoulos surged into the area and headed home captain Giorgos Karagounis' free-kick.
John Buttigieg's Malta team got back into the game nine minutes after the break when Mifsud slotted home after some sloppy defending and their substitute Andrew Cohen was unlucky when his long-range drive cannoned back off the left post.
But the game was put to bed by Fetfatzidis after 64 minutes, the talented winger slotting into an empty net after goalkeeper Andrew Hogg had parried Dimitris Salpingidis's cross.
