He added: "They were dying on the pitch, we are proud with them".
Russian Federation head coach Stanislav Cherchesov said his team were understandably disappointed with the result.
"Thank you for those honest, genuine emotions that you gave us throughout the World Cup". To beg someone to trust in you, we didn't want to do that.
Earlier players partied inside the dressing room in a raucous atmosphere as they toasted reaching the final four of the tournament. "When you've got all these other teams, France, Belgium, England, that's the top teams from the beginning and we are the underdogs".
"Honestly, I'm not even aware of what we've achieved", said the 27-year-old, who cancelled out Denis Cheryshev's spectacular opening goal with a headed finish late in the first half. "We showed how calm and self-confident we are".More news: World Cup Quarter-Final match report Russian Federation v Croatia 07 July 2018
"It's lovely", said one Brazilian fan in the St Petersburg fan zone.
"Unbelievable!. I can't believe that we have again won in the same way". In such instances, those who have kept consistently higher performances over a larger number of games were rewarded with a higher rank.
Croatia will now face England on Wednesday evening at the Luzhniki Stadium in Moscow with a place in the World Cup final at stake.
First up Les Bleus v De Rode Duivels in Saint Petersburg on Tuesday night, a match-up of western European neighbours.
Russian opposition politician Aleksei Navalny, who has served several jail sentences for organizing antigovernment protests, also hailed the team's effort.More news: Armed, barricaded suspect may be former NBA, UCLA basketball star
He wrote on Twitter: "It's unfortunate, but nonetheless it was a great game".
But after Russia's shock win over Spain on penalties in the round of 16, Yegorov found, to his delight, that he would be seeing his own national side play on Saturday. Fans spilling out of bars chanted "Russia" in the street and danced as music blared.
"It was a great match".
However, it was not to be for the resilient hosts, with goalkeeper Danijel Subasic saving Fedor Smolov's penalty and Fernandes firing wide before Rakitic kept his composure to send Croatia through.
This time, it was the Croatian fans' turn to celebrate.More news: Kazan confirmed as World Cup graveyard of champions
Russia's unlikely World Cup run has captured the imagination of the nation, with celebrations the length and breadth of the country after each of the team's wins in the tournament.