كتب : Hatem Maher
Konstantin Zyryanov scored the winner for Russia 12 minutes before the interval.
The victory revived Russia's hopes of reaching the quarter-finals as they lie third on three points. Just goal difference separates them from second-placed Sweden.
Spain, who are on top with six points, guaranteed their place in the quarter-finals while Greece suffered a premature exit following their second consecutive loss.
Roman Pavlyuchenko tried to lob Antonios Nikopolidis after 14 minutes but the Greece keeper tipped his effort over the bar.
However, the Olympiakos guardian was to blame for Russia's opener in the 33rd minute.
He needlessly rushed towards the right hand side of the penalty area, allowing Sergei Semak to acrobatically kick the ball into the path of Zyryanov for an easy tap in.
In the second half, Pavlyuchenko wasted several clear-cut chances to double Russia's advantage.
But the one-time European champions still managed to hold onto their lead to claim a precious win.