Yuri Kovtun was credited with Russia's opening goal in the 36th-minute even though his shot was deflected into the net. The Greeks rallied to equalise with a fine header by Yotis Tsalouhidis in the 64th minute, but Russia wrapped up the victory seven minutes later with a goal by Viktor Onopko.
The Republic of Ireland gained the chance to win Group Six when the leaders, Portugal, managed only a 1-1 draw with Austria in Vienna.
Spain qualified from Group Two by drawing 1-1 away to second-placed Denmark. Switzerland are on the verge of their first Championship after beating Hungary 3-0 in Group Three and Turkey drew 0-0 in Iceland.
France gave themselves a chance of winning Group One by beating the section leaders, Romania, 3-1 in Bucharest.