Greece, who upset Russia with a first-half goal from captain Giorgos Karagounis in their must-win match, went through as Group A runners-up and will play the Group B winners in the last eight.
"This result was the fruit of the efforts and decisiveness of the players on the pitch. God was with us," Santos told reporters.
"Focus and concentration decided this match. We didn't allow our opponents to exploit their strengths and score a goal... the character of the team emerged when they were in this situation and the reply was fantastic."
Karagounis, winning his 120th cap to equal the record for his country, made Russia pay for a flurry of missed chances when he scored against the run of play deep into first-half stoppage time.
Studying the Russian team, Santos said, revealed a weakness at set-pieces and their need for lots of space in order to create chances, which Greece answered with a rock-solid defensive line.
"They need a lot of space, especially in the last third of the pitch in order to create dangerous chances," he said. "So we needed to be very concentrated in not allowing them to deploy their skills and the talent that they possess."
Facing repeated questions trying to relate the fate of the Greek team to debt-strapped Greece's wider fate, Santos said: "Everyone has to respect Greece. Everything started in Greece. It is very difficult for anyone to give us lessons."
"From tomorrow we have to clear our minds and keep our feet on the ground. We know who we are, and we will give everything in the following match as well."
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