Dick Advocaat's side, who went into the match as Group A leaders, outclassed their opponents in a stadium packed with Russian fans but they struggled to get chances on target.
And in first-half stoppage time Greece captain Giorgos Karagounis, the only surviving member of their 2004 championship-winning side, grabbed the only goal of the game to give them an unlikely slot in the last eight. But victory was bittersweet for Karagounis, who earned a record-equalling 120th cap last night. He will be suspended for the quarter-final after being shown a controversial second-half yellow card shown for diving.
Russia went into the game in the dangerous position of knowing they needed only a draw to go through. It was win or bust for their opponents, bottom of the table with a single point, but Greece got off to the fast start they failed to produce in their previous games by winning a succession of early corners.
Russia goalkeeper Vyacheslav Malafeev made an acrobatic save from one of them when Kostas Katsouranis hooked Karagounis' sixth-minute delivery within his reach – although it was going wide anyway.
Vassilis Torossidis also had a shot blocked before Russia's technical superiority began to show. But Andrey Arshavin poked Alan Dzagoev's 10th-minute cross straight at Greek keeper Michalis Sifakis and Aleksandr Kerzhakov was inches wide with a superb 20-yard half-volley.
Former Chelsea winger Yuri Zhirkov spoilt a thrilling break with a wayward cutback, the final ball continuing to frustrate Advocaat's men as Greece worked their way back into the game.
Kerzhakov should have done better seven minutes from half-time when he produced an embarrassing air shot from Dzagoev's knockdown. Zhirkov also sent a stunning long-ranger inches over the top.
For Greece, lone striker Fanis Gekas was an isolated figure, with Giorgos Samaras having to track back to fortify his overworked midfield and defence as Russia probed down each flank. The half looked set to end 0-0 until Karagounis pounced on a defensive mistake before drilling the ball under Malafeev.
The pattern of the first half continued after the break with Roman Shirokov and Igor Denisov off target. Greece were convinced they should have been awarded a penalty when Karagounis tumbled as Sergei Ignashevich stuck out a leg.
Instead, Karagounis was booked for diving, much to his dismay and that of coach Fernando Santo, who held his hands to his head on the touchline.
Greece defender Giorgos Tzavellas crashed a free-kick against the angle of post and upright in a rare Greek attack in the second half.
Russian fans had poured into the Polish capital during the day and heavily outnumbered their Greek counterparts, turning Warsaw's National Stadium, which was nearly full to capacity, almost into a "home" venue for Advocaat's side. But there were no reports of a repeat of the street violence that marred the build-up to Russia's clash with Poland on Tuesday.