"President Putin called me before the game with the Spanish and after," Cherchesov said at a press conference in Sochi on Friday. "Of course when the president supports you, it makes us comfortable.
"The players know this and it is just an extra boost for us in terms of motivation."
Putin watched on television as the hosts stunned 2010 World Cup winners Spain on penalties at the weekend to secure Russia's place in the quarter-finals for the first time since the Soviet era.
There was euphoria across the country following the victory, as a Russian team written off before the tournament even began continued to defy expectations.
Cherchesov's team came into the World Cup without a win in seven games, and the coach is keeping his feet on the ground ahead of Saturday's match at the Fisht Stadium on Russia's Black Sea coast.
"Myself as a coach, openly, I'm trying not to watch television or read the newspapers. I'm concentrating on my job and that's it," said the 54-year-old.
"I believe the players are the same. It is good to remind ourselves of the game against Spain, but let's look forward now. This is a different game now, a different level in the quarter-final."
Cherchesov has also been boosted ahead of the Croatia game by the return of Alan Dzagoev.
The CSKA Moscow attacking midfielder has not featured since coming off with a hamstring injury in the first half of the opening match, a 5-0 drubbing of Saudi Arabia in Moscow on June 14.
"It was a pity to lose such a player in the first game. He is already training... and we will see If I can play him from the first minute," said the coach.