Greece remain rooted to the bottom of Group F after the 2004 champions lost their Euro 2016 qualifier against Finland.
Joel Pohjanpalo's goal 15 minutes from time at the Stadio Georgios Karaiskakis also means that Greece are four points adrift of the Faroe Islands and have no mathematical chance of reaching next year's finals.
The hosts had started brightly, with Kyriakos Papadopoulos heading wide inside the opening six minutes.
Soon after, Konstantinos Fortounis fired another effort narrowly wide of Lukas Hradecky's goal as Greece tried to make their early pressure count.
Nikolaos Karelis then tried his luck with a strike from distance, which Hradecky was able to save comfortably.
Greece's inability to break through the Finnish backline was punished when the visitors took the lead on 75 minutes after Pohjanpalo rifled the ball into the top corner of Orestis Karnezis's net.
The home side tried to force a late equaliser, but both Panagiotis Tachtsidis and Jose Holebas were denied by Hradecky as Finland held on to move into fourth.
