"[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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