"I'm not concerned with who we play next, it's just great to have done this," said Greek striker Georgios Samaras.
Russia had been the overwhelming favorite going into the match at the National Stadium, but it was again guilty of squandering chances. The Russians finished with 25 shots on goal, while Greece managed only five.
"We should have won by a wide margin tonight but we didn't. My compliments to Greece," Russian coach Dick Advocaat said. "A number of players weren't sharp enough to score."
The Russians became one of the tournament favorites with their 4-1 win over the Czech Republic in their opening Group A match, but then drew with Poland and finally lost to Greece, snapping a 16-match unbeaten run.
Russian players dejectedly tramped off the field immediately after the final whistle, while Greece's euphoric players celebrated with fans.
"What I'm feeling is overwhelming. I think we did deserve more, but it didn't happen," Malafeev said. "To concede that goal just before halftime was an awful feeling."
Czechs oust Poland
In the other Group A match, in Wroclaw, Poland, Petr Jiracek scored a second-half winner to give the Czech Republic a 1-0 victory over Poland, and a place in the quarterfinals.
Tournament cohost Poland was knocked out of the tournament while the Czechs finish as Group A champions.
Pushing forward in the 72d minute, Jiracek collected a pass from Milan Baros in the box, cut right past Poland's Marcin Wasilewski, and slotted the ball inside the far post. Poland, which had to win to advance, immediately brought on two attacking players in the last 30 minutes to push for a goal, but struggled to create any serious threats.
After the final whistle, the Czech players swarmed each other, while the Poles fell to the ground, covering their faces.
U.S. women beat Sweden
Abby Wambach's 136th career goal for the U.S. women's team led the Americans past host Sweden, 3-1, in Halmstad in a tune-up for the London Olympics.
Wambach connected just 8 minutes in, then Alex Morgan scored in the 22d. Sweden's Lotta Schelin scored in the 35th minute, and substitute Tobin Heath sealed the game with a header 11 minutes into the second half.
Wambach's goal came off a punt from goalkeeper Hope Solo. Heather O'Reilly won the ball in the air, nodding it to Wambach. She turned on a defender and her half-volley from 35 yards caught keeper Hedvig Lindahl out of position.
Le Toux sparks Vancouver
Sebastien Le Toux's goal in the 80th-minute gave the undermanned Vancouver Whitecaps a 1-0 victory over the visiting Colorado Rapids.
Le Toux scored on a sliding volley off Davide Chiumiento's long cross from near midfield. Chiumiento was the key to the play as he dribbled away from three or four defenders in his vicinity.
Vancouver played the last 30 minutes a man down after striker Darren Mattocks was ejected in the 60th minute following his second yellow card.
Crew, Revolution draw
Andy Gruenebaum made five saves for his fifth shutout of the season to lead the Columbus Crew in a scoreless draw against the host New England Revolution.
Houston downs Dallas
Adam Moffat scored his first goal of the season in the 76th minute to give the host Houston Dynamo a 2-1 win over FC Dallas.