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.
Calcutta/Sochi: Croatia beat Russian Federation 4-3 via penalty shootout in Saturday's last quarterfinal tie, at the Fisht Stadium in Sochi, to advance to their second World Cup semi-final after 1998.
The game finished 2-2 after extra-time following a late equaliser by Russia's Mario Fernandes.
Croatia won 4-3 in a penalty shootout after the quarterfinal was tied 1-1 after regulation and 2-2 after extra time.
The hosts qualified for the last eight after a dramatic penalty shoot-out victory over former world champions, Spain.
They return to action on Wednesday when they meet England (19:00 BST) in Moscow for a place in the final on 15 July.
He added: "They were dying on the pitch, we are proud with them".
"They fought. It was a real playoff match against not the weakest team in the world".
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"We have already achieved something big by reaching the semi-finals".
"But they [Croatia] have been behind in both knockout games and then they had the setback of missing a penalty in extra-time against Denmark and still found a way on penalties".
"After the game, Putin phoned him again and noted the team's worthy performance at the World Cup, as well as Russia's lovely and determined game in the historic match against Croatia".
It was arguably France's most impressive display at the World Cup.
Ivan Rakitic scored the victor from the spot as Russian goalkeeper Igor Akinfeev dived the opposite direction. I asked him and he said he was OK, it did not affect him much.
Mr Peskov said: "He watched, he was rooting for the team".
The second half gave us very little but Ivan Perisic looked certain to score when he shot from inside the area, the ball hit the inside of the post but somehow didn't go in.
Russian Federation midfielder Denis Cheryshev scored the opener in the 31st minute with a stunning left-footed strike from distance, to notch up his fourth goal of the tournament.
Conseillé:
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The game went to extra-time with Subasic staying on the pitch despite appearing to injure his hamstring late on in normal time. Captain Luka Modric is expected to partner Ivan Rakitic in the midfield, while Mandzukic will lead the attack line.