The Greeks soaked up pressure throughout the match before exploiting a defensive lapse from the Russians which proved to be an expensive mistake
Despite being under pressure for large parts of the game the Greeks took the lead in first-half stoppage time as Karagounis punished some sloppy defending with a neat finish.
And his side remained resolute in defence throughout the second half to secure a famous victory which, coupled with the Czech Republic's win over Poland, meant Russia finished third in Group B.
The Ethniki came into the game knowing they needed a win to ensure qualification and they were the first to draw a save from the opposing keeper.
|MATCH FACTS | Greece 1-0 Russia
Russia responded positively, and three minutes later Alan Dzagoev broke in from the right following a quick throw. Andrey Arshavin got on the end of his cross 10 yards from goal and Michalis Sifakis was forced to parry the ball away.
Dick Advocaat's side were in the ascendancy and moments later had two great chances to go ahead. Firstly, Aleksandr Kerzhakov failed in his quest to get his first shot of the tournament on target as his volley bent agonisingly wide of the far post. Then, Yuri Zhirkov's cutback for Dzagoev was intercepted by Katsouranis.
By the half-hour mark the Russians had taken control of the match but were more often than not being restricted to shooting from range. Yet, when in possession, the Greeks lacked ambition and committed minimal numbers to their attacks.
Zhirkov fired over five minutes before the break but Fernando Santos' side held out and in stoppage time Karagounis fired Greece into a surprise lead. A misplaced header by Sergey Ignashevich was capitalised on by the 35-year-old who capped his 120th international appearance - a statistic equalling the country's all-time record - by surging into the area and slamming a low shot underneath Malafeev.
Roman Pavlyuchenko replaced the profligate Kerzhakov at half-time but the Russians continued to struggle against the resilient Ethniki defence. Their woes were best summed up when, with space limited in the final third, Igor Denisov shot from distance only to see his effort swing wide.
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Greece remained a threat on the counterattack and after Aleksandr Anyukov somehow cleared Vasilis Torosidis' cross away from inside the six yard box, Karagounis received a booking that will keep him out of the next match for diving in the area though it looked like a certain penalty.
Santos signalled his intent to see out the game with the solitary goal by replacing striker Theofanis Gekas with left-back Jose Holebas 25 minutes from time. But the Ethniki almost extended their lead in the 70th minute when Gergios Tzavellas' beautifully struck free kick rattled the frame of the goal.
In their desperation to find an equaliser Advocaat's side became extremely disjointed and another long-range attempt from Denisov was saved by Sifakis. With five minutes of normal time remaining Dzagoev came close to restoring parity but his header from Arshavin's cross shaved the outside of the post.
And the Greeks held out to secure their passage to the quarter-finals whilst Russia - who had shown so much promise in their opening game against the Czech Republic - face an early flight home.