Sweden will be slight favourites to advance as they only need a draw in Innsbruck on Wednesday to go through from the group stages - but Russia's confidence will be high after they rediscovered the form that led to them edging out England for a place in the tournament.
Greece, in contrast, were unable to repeat the heroics that stunned Europe and resulted in them clinching the title in Portugal four years ago - paying a heavy price for a lack of cutting edge and attacking ambition.
Spain are the group winners and now have the opportunity to rest key players when they face Greece in their last match before the quarter-finals.
The defending champions needed to show greater positive intent than they produced in a sterile show when they lost to Sweden but it was Russia who threatened first when keeper Nikopolidis turned a clever chip from Pavlyuchenko over the bar.
Charisteas was Greece's hero when they lifted this crown four years ago but he was guilty of an awful miss when presented with a perfect chance after 19 minutes.
Angelos Basinas's free-kick invited a simple headed finish but Charisteas barely got a touch and the ball rebounded to safety off Igor Semshov.
And Russia were grateful for the reprieve as they took the lead after 34 minutes, thanks in no small measure to a dreadful piece of goalkeeping from Nikopolidis.
He inexplicably chased Diniyar Bilyaletdinov's cross beyond his far post, giving Semak the opportunity to hook the ball back into the area for Zyryanov to apply the simplest of finishes.
It was the start of a spell of Russian supremacy, with Bilyaletdinov firing just over the top from 25 yards.
Greece were struggling to get a foothold in the game and they suffered another blow after 39 minutes when Giourkas Seitaridis limped off to be replaced by Giorgios Karagounis.
Charisteas was having a desperate evening and he was the culprit again seconds after the break, lobbing a lame finish straight at Russian keeper Igor Akinfeyev from a Karagounis pass.
Russian were soon back on top, with Pavlyuchenko twice forcing saves out of Nikopolidis before shooting into the side-netting after a surging run.
Karagounis then forced an instinctive block out of Akinfeyev as Greece attempted to keep themselves in Euro 2008.
Greece were enjoying their best spell of the game and Basinas should have done better than shoot over the bar when unmarked as Russia suddenly came under pressure.
It was an absorbing contest and Bilyaletdinov pulled a finish just off-target after a clever flick from the dangerous Pavlyuchenko.
Charisteas's luck was out again after 67 minutes when he headed another presentable chance straight into the arms of Akinfeyev as the hero of 2004 became Greece's villain.
Yuri Zhirkov almost clinched the win for Russia with an angled free-kick that drifted just wide of the target, with Nikopolidis struggling.
Pavlyuchenko's excellent all-round game had been undermined by his wayward finishing and he blazed another great opening over the bar in the dying seconds but there was no time for Greece to mount a late recovery.
Next to hosts Switzerland, Greece are the second country to be knocked out of Euro 2008. Greece are the fourth defending champion not to have made it through the group phase.
Greece are the first defending champion to lose and fail to score in their opening two matches, and the only country that has not scored at Euro 2008. The last country that failed to score at a European Championship edition was Denmark in 2000.
Giorgios Karagounis received his fifth career yellow card at the European Championship. He has now equalled the record for most yellow cards set by Gheorghe Hagi in 2000, Pavel Nedved in 2004 and Gianluca Zambrotta in 2008.
Russia can become the eighth country to reach the knock-out phase despite losing the first match. They could also be the first to do so having lost the first match by a three goal margin.
Due to Russia's win, Spain have qualified for the quarter-finals as group winners. Russia need a win against Sweden to reach the last eight.
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