Fedor Smolov, so often the darling of Russian football, stepped up to take the first penalty but his underhit shot was saved by Subasic to immediately put the hosts on the back foot.
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.
It promises to be a fascinating encounter Sweden & England battle it out for a place in the semi-finals of the World Cup. Both Luca Modric & Ivan Rakitic didn't see enough of the ball against Denmark, and the Croatia boss may instruct his star midfielders to drop slightly deeper to start to dictate the play. The hosts came into the tournament as the lowest-ranked team but began with a bang by beating Saudi Arabia 5-0 and Egypt 3-1. That strike was canceled out eight minutes later by a Andrej Kramaric header. Andreij Kramaric then equalised for Croatia and it remained like that till the end of the second half.
It was a match that, frankly, was pretty boring for the first 30 minutes.
Croatia were agonisingly close to scoring on the hour when Russia's defence failed to clear the ball and Ivan Perisic's shot hit the inside of the post but rebounded harmlessly across the face of the goal.
Perisic glanced a header just wide of the left post from Vrsaljko's fine cross on the right in the 28th minute and it seemed only a matter of time before the Croatians scored.
Back up the other end, Mandzukic directed a header over from a 77th-minute corner in what was the final attempt in regulation time, but the first extra-time goal of the World Cup arrived on 100 minutes and it was Croatian fans celebrating.
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.
Russian Federation became the third World Cup hosts to have their tournament end on penalties, after Mexico in the 1986 quarterfinals and Italy in the 1990 semifinals.
Croatia did finally go ahead when Vida met a corner with a weak downward header which found its way through two sets of legs and rolled into the net.
Croatia was a flawless three-for-three on their final penalty kicks, and Ivan Rakitic converted the game-winner to advance Croatia.
In the final minutes, Subasic, Croatia's own last 16 penalty shootout hero, collected the ball and writhed around in apparent agony.
Subasic was forced to make one more save to deny Roman Zobnin, and the game went to penalties.
The Villarreal winger replaces Yuri Zhirkov, who has been struggling with a calf injury, in the only change made by coach Stanislav Cherchesov following the penalty shootout win over Spain in the last 16.