After a bright start the Czechs faded badly as Alan Dzagoev and Roman Shirokov put Russia 2-0 up at the interval.
Vaclav Pilar did pull one back but Dazgoev's second and a solo effort from substitute Roman Pavlyuchenko condemned the Czechs to a heavy defeat that could have been worse had it not been for some woeful Russian finishing.
While the result puts a dent in Czech hopes of reaching the last eight, Bilek insists qualification is not beyond his side if they can pull together ahead of Tuesday's meeting with Greece.
He said: "I think we started well, in the first 15 minutes we had plenty of possession, but there were some attacking situations we did not finish off. Then we lost the ball in our opponents' half, they broke quickly and it was a mistake that cost us.
"We tried to change it after half-time and we scored a goal but then we lost the ball again and the opponent punished us and that is why the result is what it is. We had problems with the combinations in Russia's midfield and that is how they had so many chances.
"We are very disappointed but now we have to show our strength and our team spirit. We are still able to win and the players have a chance to perform in the following two matches so we are going to look forward now."
Advocaat calls for more
Russia coach Dick Advocaat declared himself pleased with his side's impressive victory, but warned their finishing must improve if they are to have any chance of winning the tournament.
The Dutchman said: "We are quite happy, we scored four goals and in any international game that is a very good result. We could have scored more goals, in fact we should have scored more as we had a lot of chances.
"At 2-1 we made it very difficult for ourselves so we have to get better at that because maybe a better team would have scored more against us.
"It could have been 2-2, there were moments where we need to be sharper, at certain points they had a lot of space and were very dangerous. But this was our first game and we are really happy with the result."
Advocaat reserved particular praise for Andrey Arshavin, who is currently on loan at Zenit St Petersburg from Arsenal, after the midfielder inspired his side with a masterful display.
He said: "He worked very hard and he played very well. He is a very important player for us for the way we play and he was very, very good. I am very positive about him."