Croatia, who reached their first semifinal since 1998, dominated possession over the 120 minutes but were let down by poor final passes and were vulnerable to Russia's aerial game.
Russian Federation had an opportunity when Dejan Lovren gave the ball away in his own third, but was able to recover to muscle Fedor Smolov off the ball.
Cheryshev exploited it in a way the likes of Jesse Lingard and Harry Kane could come Wednesday, launching a long-range effort which had Danijel Subasic helpless in the Croatia goal.
The crowd erupted, but Croatia were undeterred and equalised just eight minutes later.
The arrival of the host nation for the final World Cup match at the Fisht Stadium meant by far the most vibrant atmosphere yet at this Black Sea beach resort.
The game went to extra-time with Subasic staying on the pitch despite appearing to injure his hamstring late on in normal time.
But Pivaric was whistled for handling the ball at the corner of the box, and Dzagoev's cross was flicked inside the back post by Fernandes to level the game in the 115th minute.
Zlatko Dalic's charges nearly came unstuck though in the last 16 round against a stubborn Denmark side in a match they eventually won 3-2 on penalties, following a 1-1 stalemate after 120 minutes. Striker Mario Mandzukic ran unchallenged into the penalty area and chipped a low ball back into the centre where the unmarked Kramaric scored with a glancing header.
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Captain Luka Modric is expected to partner Ivan Rakitic in the midfield, while Mandzukic will lead the attack line.
And against Croatia came a third, the likely victor, as Cheryshev one-twoed with Artem Dzyuba down the left before curling powerfully into the top left corner from 25 yards, giving Russian Federation the lead.
Croatia went agonisingly close to taking the lead for the first time on the hour, but Perisic was denied by the upright after Igor Akinfeev flapped at a lofted ball from Kramaric.
Domagoj Vida looked like he had won it for Croatia in extra-time, only for naturalised hero Mario Fernandes to find an even later equaliser to force a penalty shootout.
Vida removed his shirt as he ran away in celebration, possibly thinking Croatia were heading through.
Kramaric threatened again seven minutes into the second half, although his overhead-kick was comfortably claimed by Akinfeev, who saw Ivan Strinic fire well over the crossbar from 22 yards out on the left in the 58th minute, before the woodwork came to Russia's rescue two minutes later.
There is a comprehensive argument that Luca Modric has been the best player at the World Cup.
- 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.
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