The home team punished them when Cheryshev came in from the left and, after a one-two with Artem Dzyuba, skipped away from Modric to curl an exquisite 31st-minute effort beyond Subasic, who remained rooted to the spot as the ball sailed in.
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.
Danijel Subasic was rooted to his spot, although quite why he was stood on the edge of his six-yard box as Cheryshev advanced is beyond us. He was as stunned as everybody else who witnessed it.
Russian coach Stanislav Cherchesov goaded the crowd to ramp up the volume and he was duly obliged as the noise became deafening during penalties.
Cherchesov's men were unable to hold on until half-time, though, as Mario Mandzukic advanced down the left to set up the chance for an unmarked Kramaric to head home and silence Sochi.
Russian Federation substitute Alexander Erokhin headed over the bar and Modric had a shot blocked before referee Sandro Ricci's whistle signalled the game was heading into extra-time. This time, it was Prime Minister Dimitry Medvedev's turn to shake his opponent's hands and shrug.
His first, in the opening 5-0 win over Saudi Arabia would most likely finish third on this list for its simpler finish, but Cheryshev had the most lovely of first touches, delicately flicking it up over the sliding defender, taking another touch before firing it home and doubling Russia's lead. Be it "Ole, Ole" or the Icelandic thunderclap or even the Seven Nation Army, the host nation had borrowed and learned the famous chants and celebrations. The team had made them believe.
Perhaps Russia might have benefited from having Cheryshev available when a long, sapping evening ended in a penalty shoot-out. However, Croatia played with much confidence and initiative. As the results of a match of Russia and Spain this tactic for the Russians is very suitable to help you reach decide the championship.
Then came the real drama. Nobody was questioning their abilities today. Croatia began warming up its backup goalkeepers in case Subasic was unable to continue into extra time. But they weren't making mindless passes. In Russia's next two attempts, Sergei Ignasevich and Kuzyayev were successful but as did Vida and Ivan Rakitic to give Croatia a 4-3 triumph.
After earning a corner himself, centre-back Vida got on the end of Modric's cross to head it down and somehow through two Russian defenders and Corluka into the corner. Their roller coaster ride had just taken a dip. Just when it seemed like the game was lost, Mario Fernandes's equaliser sent everyone into raptures.
In the end, though, it wasn't to be.