That meant penalties would decide the match, with Croatia running out winners and setting up a date with England at Luzhniki Stadium in Moscow.
Russian Federation struck first with a 25-yard goal from Denis Cheryshev in the 31 minute.
Mario Mandzukic, who had missed a sitter in the opening minutes of the game, sent a ideal assist from the left for Andrej Kramaric to creep up on two defenders and head home.
Brazil-born Mario Fernandes had kept Russia's dream alive when he headed home in the 115th minute to level and force penalties.But he and Fyodor Smolov both failed to convert their spot-kicks in Sochi. While it was never a classic, this game will be warmly remembered by the Croatians for their heroic display as they ground out a 2-2 draw to win 4-3 in the penalty shoot-out.
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. Assisted by Alan Dzagoev with a cross following a set piece situation.
Croatia continued to dominate possession in Russia's half, though the hosts had a rare chance in the 72nd minute as Aleksandr Yerokhin found space in the box but sent his header over the bar. Domagoj Vida jumped the highest to head the ball in, and Croatia stunned the crowd to silence. Luka Modric, who was yet again the best player on the pitch for Croatia, sent shivers down the spines of his compatriots when his penalty hit the bar, but it had sneaked in past Akinfeev.
39 - Goal! Russia 1, Croatia 1. But Croatia made it 1-1 just eight minutes later thanks to a headed goal from striker Andrej Kramaric. Assisted by Mario Mandzukic.
As time ticked away, the side that hammered Argentina 3-0 in the group stage seemed to be running out of ideas, perhaps the biggest compliment to Russia's tactical discipline.
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Zverev was once again taken to five sets for the fifth time in a month but he fell short against the world No 138. Kontaveit won with a score of 6-2, 7-6 (4). "I was feeling physically really good".
To a home semi-final Russian Federation on the 32nd minute brought Cheryshev, who managed a great shot into the far top corner Subasic.
Certainly, every time the ball got into the final third, the whole stadium was willing it to nestle in Danijel Subasic's goal. Both sides went all the way to penalties in the round of sixteen and will be looking to get the job done in 90 minutes to avoid fatigue.
Artem Dzyuba had a volley blocked by Dejan Lovren as Russian Federation looked to make an early mark on the game, and an Ante Rebic header and Rakitic free-kick looped over the bar at the other end.
After the restart, play resumed much as it had begun in the first half.
Even as Croatia tried to take control of the game once again, Russian Federation were given chances to break on the counter-attack, something England can take advantage of.
But it remains to be seen how quickly the team - with President Kolinda Grabar-Kitarovic watching from the stands - can recover from their efforts, having also been taken to extra time and penalties by Denmark in the last round.
Croatia will now look to go one step further than the great side of 20 years ago, who lost to France in the last 4.