A rare goal from Diane Caldwell was enough to give the Republic of Ireland senior side the win they needed to stay in contention for qualification for Euro 2021.
A close-range finish from Caldwell three minutes before the half-time break was the only difference between Ireland and Greece at Tallaght Stadium.
Ireland had plenty of possession and chances but not enough end product to inflict more damage on the Greeks, but the main thing for coach Vera Pauw to take away was the win and a second clean sheet in four qualifiers.
The win puts Ireland well clear in second place, six points ahead of Greece and only two points behind table-toppers Germany, so an Irish win in Montenegro on Wednesday would even dislodge the Germans, temporarily, though they are expected to win the group leaving Ireland hoping for a place in the playoffs.
The Irish side had been unable to get past Greece's keeper before Caldwell scored on 45 minutes.
A Katie McCabe free kick was nodded on by Louise Quinn and Caldwell was left unmarked in the box to score only her second international goal.
Second half chances from Ruesha Littlejohn, Aine O'Gorman and sub Amber Barrett could not add to the goal tally and the one goal had to be enough for the Irish.