"The president has been calling me, and that's been an extra boost for us in terms of motivation", Cherchesov reported, as he and his players arrived at the Black-Sea coast for the quarter-final against Croatia.
Having edged past Denmark on penalties in the last 16, Croatia overcame hosts Russian Federation in another dramatic shootout.
Croatia grabbed the lead in extra-time thanks to a header from Domagoj Vida following a corner, but with just six minutes to go Russian Federation made it 2-2 when Mario Fernandes glanced home a header following a free-kick.
Denis Cheryshev scored a brilliant opener in the first half but just eight minutes later former Leicester City striker Andrej Kramarić headed the Croatians level.
It was a dramatic end to the match for Subasic who went down with what appeared to be a hamstring injury late in regulation time.
Fernandes, who was born in Brazil but rejected a chance to play for that country's national team in 2011, sent his penalty kick wide of the net in the shootout, giving Croatia the advantage.
"This was not a attractive game, it was a fight, a battle for the semi-final", added Dalic, who broke down in tears after Croatia confirmed a first World Cup semi-final appearance since 1998. Igor Akinfeev later blocked an attempt from Mateo Kovacic.
Rakitic then scored his second penalty shoot-out victor in two games. The English team defeated Sweden 2-0. This is the first time two teams have had penalties in successive matches of the knockout stages of a World Cup, though.
Ivan Rakitic, so often a key man for Barcelona, was not almost as influential as at club level, while in defense, Russia's Mario Fernandes beat more highly rated defenders to head home a late equalizer.