MOSCOW -- Russian President Vladimir Putin is "proud" of the national team, Interfax news agency cited a Kremlin spokesman as saying, after Russia lost to Croatia 4-3 on penalties in a World Cup quarterfinal match on Saturday.

"[Putin] watched, he was rooting for the team. We lost in a fair and great game. They are still great guys for us, they are heroes. They were dying on the pitch, we are proud with them," Kremlin spokesman Dmitry Peskov was quoted as saying.

Croatia will play England, who earlier beat Sweden 2-0 in Samara, at the Luzhniki Stadium in Moscow on Wednesday.

Russia head coach Stanislav Cherchesov said his team were understandably disappointed with the result.

"I believe that the keyword is how we leave -- it's better to leave when you feel pride," Cherchesov told reporters. "We don't have any chances to win this Cup and of course we are sad. But we showed our worth."

Fyodor Smolov had his first penalty saved by Danijel Subasic and, although Russia goalkeeper Igor Akinfeev saved a Mateo Kovacic kick, their fate appeared sealed when Mario Fernandes fired wide from their third attempt.

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Cherchesov said the hosts, who were ranked 70th by FIFA before the World Cup, should aim higher.

"From the first second I put together this team, I knew where it was heading," the coach added. "We have succeeded I believe, but it's time to take a step forward."

Cherchesov said Putin had called him before the match and after.

"Vladimir Putin talked to me during the day and he called me right now, and congratulated me on a very good game," Cherchesov said. "We told him we were disappointed. He said that we should keep our eyes open and take the next steps."

Meanwhile, Russian football fans saluted their team after the underdog host side lost on penalties to Croatia, bringing an end to their World Cup challenge at the quarterfinals.

Across Russia, hand-wringing fans had held their breath through extra time, hoping the team could pull off another upset in a World Cup they entered as the lowest ranking side, but which saw them reach the quarter finals against the odds.

Russia fans react after their national team was eliminated from the 2018 World Cup by Croatia. Getty Images

The disappointment of defeat late on Saturday soon melted into applause as fans toasted a national side that had repeatedly defied expectations. Fans spilling out of bars chanted "Russia" in the street and danced as music blared.

"Our boys, they really did great. A huge thank you to them for this tournament. What we achieved, that was so cool," Andrey, a lawyer, said next to a street corner screen that had shown the match.

"It was a great match. Well done to our guys, they tried very hard... I'm really happy that we made it to the quarters for the first time in history," said Artyom Osadchy, a student.

Other fans looked to the future on the heels of Russia's best World Cup performance since the collapse of the Soviet Union.

"You can't take the trophy straight away. It's the first time we're in the quarter finals in a very long time. You can't have everything straight away. It was a very dignified game," Evgenia, 36, a psychologist, said.