Just a week ago they were being hailed as possibly the best team in the tournament and Advocaat said Russia had let the prize of a quarter-final slip out of their hands after sitting top of Group A after the first two matches.
"Even though we lost today we played brilliantly in the first half," he told reporters. "Then they got a goal and that created a different story."
Russia had numerous chances to pull ahead early on but Greece captain Giorgos Karagounis netted two minutes into first-half stoppage time.
"We didn't have the sharpness to score that goal," Advocaat said.
Russia had not lost any of their previous 16 games and a draw would have put them through.
"We're there to win the game, not to draw. We tried to control the game and we tried to decide the game," Advocaat said.
The Dutchman also complimented the Greek defence that managed to stymie the Russian pressure: "The other team just defended well."
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