Fernandes then went from hero to villain, the scorer of the equalizing extra-time goal dragging his penalty kick woefully wide, handing Croatia the advantage.
Crotia then went in front with a soft header from substitute Domagoj Vida in the 101st minute, the first goal scored in extra time at the tournament. Both teams are coming into the World Cup with thrilling victories in the quarter finals against Spain and Denmark respectively.
"We just wanted to prove ourselves", said coach Stanislav Cherchesov.
Andrej Kramaric and Domagoj Vida got the goals for Croatia, while Denis Cheryshev and Mario Fernandes were on target for Russian Federation.
After Akinfeev fluffed his punch in an attempt to clear a cross, Perisic struck the inside of the left post from eight yards out and watched the ball cannon past the opposite upright.
With the World Cup starting to come to a close, both sides have staked a legit claim as to why they should be moving onto the final four, but while Russian Federation shocking Spain was a huge achievement for the side, Croatia will be a different task altogether.
Croatia's Andrej Kramaric, right, scores his side's first goal during the quarterfinal match between Russia and Croatia at the 2018 soccer World Cup at the Fisht Stadium in Sochi, Russia, Saturday, July 7, 2018.
Croatia goalkeeper Danijel Subasic once again came up big, making some big saves in regulation, one after getting injured, and saving Russia's first penalty kick.
Centre-backs then were successful in the fourth round, Vida answering after Russia's Sergei Ignashevich found the net, and though Daler Kuzyaev kept Russian Federation alive, Rakitic - who also won the shootout against Denmark in the last round - put Croatia through.
The victor of today's clash will face either England or Sweden in the semi-finals.
Both sides book their places in the last eight courtesy of dramatic shoot-out wins, with Croatia beating Denmark and the hosts shocking Spain.
Subasic saved one while another went off target, and Croatia celebrated a 4-3 penalty win over the crestfallen hosts and will meet England in the semifinals next week.
"This was not a attractive game, but it was a fight", Croatia coach Zlatko Dalic told reporters. This was not a lovely game, it was a fight, a battle for the semifinal.
"We have two matches to play, we are very motivated, we will give our all". Most certainly a candidate for the best goal of this World Cup.
Croatia pressed for a late victor, earning a pair of corners, but the game went to extra-time where after four minutes, right-back Sime Vrsaljko picked up an injury and Vedran Corluka became Croatia's fourth sub.