With the crowd silenced following an extra-time header from Croatia defender Domagoj Vida in the 101st minute, Russian Federation defender Mario Fernandes scored with his own header in the 115th to send the match to yet another penalty shootout.
It was Cheryshev's fourth goal in his five matches of this World Cup. Croatia began warming up its backup goalkeepers in case Subasic was unable to continue into extra time.
The Croats hadn't advanced to the semifinals at the World Cup since 1998, when the country made its first appearance.
- Russian Federation are the first host country to be eliminated at the quarter-final stage of a World Cup since Mexico in 1986.
Their first victory came over Denmark.
Yuri Gazinskiy became Russia's third substitutes, coming in for Dzyuba, but Croatia almost ran into trouble after making two more subs, Josip Pivaric and Mateo Kovacic on for Ivan Strinic and Kramaric. Andrej Kramaric, the Hoffenheim forward, comes in with Inter Milan midfielder Marcelo Brozovic dropping out.
After a decent start, Croatia began only to keep the ball in the middle while trying to slow down the game. They got their equaliser eight minutes later in the 39th minute.
In the shootout, Smolov's feeble shot was saved by Subasic and Fernandes hit wide.
Cherchesov's men were then punished for sloppy defending as Kramaric levelled, connecting with Mario Mandzukic's cross and heading past Igor Akinfeev while the Russian defence did little to stop him.
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In 2002, 2006 and 2014 they were eliminated in the group stages, and they failed to qualify for the 2010 tournament. The last time the sides met, in a friendly in June 2015, ended 4-3 to Belgium with Marouane Fellaini scoring twice.
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Croatia national soccer team players celebrate after a penalty is saved in a shootout during the quarterfinal match between Russia and Croatia at the 2018 soccer World Cup in the Fisht Stadium, in Sochi, Russia, Saturday, July 7, 2018.
Later, Madzukic and Aleksandr Yerokhin had their chances with headers from close range as the two teams seemed to lack creativity and went after aerial movements.
The game stayed locked at 1-1 through the final whistle, prompting two periods of extra time.
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.
They will have to attack Modric earlier than they managed to do against Spain's playmakers and will probably play a far more expansive game than last Sunday.
31 - Goal! Russia 1, Croatia 0.
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.
They have not had much luck previously, however, with a 2-1 defeat against France in the semifinals in 1998 being followed by a 3-1 loss to Brazil in the Opening Match of the 2014 edition.