Liverpool's No.6 played the full game in what was a gripping quarter-final tie.
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 is only the second time that Croatia have reached the World Cup semi-finals since becoming an independent nation.
Croatia will next play England in the semifinals on Wednesday in Moscow. "I'm really happy that we made it to the quarters for the first time in history", said Artyom Osadchy, a student. As of now, only five World Cups have produced more goals in the knockout stages, with Switzerland (1954) and United States (1994) both producing 44 goals each.
"We saw how good they are from dead balls so will have to focus more on defending set-pieces because we conceded from a set-piece today, so have to improve that element of our game". We've succeeded but I believe we can make further steps forward.
Croatia coach Zlatko Dalic said Subasic's uncertain health affected how he wanted to manage his substitutes going into extra time.
They will have to play much better, " said Reuters ahead of the game.
Croatia then went ahead with a soft Domagoj Vida header in the eleventh minute of extra time but Russian Federation still managed to bounce back through Fernandes. "When you've got all these other teams, France, Belgium, England, that's the top teams from the beginning and we are the underdogs".
The men in red exceeded the wildest expectations by coming within a penalty kick and a save against Croatia of making it into the semi-finals for the first time since 1966. - Only one of the last six World Cup matches to go to extra-time has not gone to penalties - the 2014 final between Germany and Argentina.
From Brazil's squad in Russian Federation, 10 players will be 30 or younger by the tournament in Qatar in four years, including Neymar and Philippe Coutinho, who to many was the team's best player at this World Cup.
Dalic also explained why he had been overcome with emotion after Ivan Rakitic sent Croatia through.
That suggests it has had difficulty putting away its opponents, but it also means it is excelling during the biggest moments.