After ending the home side's incredible run, Zlatko Dalic's side will aim to at least go one better than the famous 1998 vintage, who finished third in France, when they meet England at the Luzhniki Stadium on Wednesday.
The hosts have surprised everyone at this World Cup and will look to continue that.
Denis Cheryshev got Russian Federation off to a brilliant start by marking his return to the starting line-up with yet another contender for goal of the tournament after his stunning efforts against Saudi Arabia and Egypt in the group stage, but Andrej Kramaric equalised just eight minutes later.
In the final minutes, Subasic, Croatia's own last 16 penalty shootout hero, collected the ball and writhed around in apparent agony. The hosts, meanwhile, have already punched much above their weight and will hope the partisan crowd at the Fisht Stadium by the Black Sea will carry them even further.
Despite the huge home support in Sochi, Croatia came into the game as favourites. Russia's success will depend on how they manage to rein in the likes of Modric and Barcelona midfielder Ivan Rakitic, and then hit on the counter as they have done with such surprising success at the tournament.
He cut in from the left, exchanged passes with Artem Dzyuba and skipped away from a Modric challenge before firing home a superb shot from 25 yards on his left foot.
As usual, Russian Federation went down fighting in front of a raucous crowd as Mario Fernandes headed a dramatic equaliser with five minutes of extra time left, only for the Brazilian-born player to turn villain as he missed their third shootout kick.
Croatia started to control possession but as they did so well against Spain, Russia was well organised defensively and gave Croatia few openings in the first half an hour.More news: Trump plans to hold talks with Putin "tete-a-tete"
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Croatia pressed for a late victor, earning a pair of corners, but the game went to extra-time where after four minutes, right-back Sime Vrsaljko picked up an injury and Vedran Corluka became Croatia's fourth sub.
Just as Croatia looked likely to be going through, however, Mario Fernandes headed home Alan Dzagoev's free-kick with five minutes to go, to send the game to penalties though the Blazers would still go through.
Croatia, a shadow of the team which cruised through their group with maximum points, went 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.
But Russia was following the visitors inch by inch as Ignashevich levelled the scores again 2-2.
Penalties were required and after Smolov's feeble Panenka attempt with the first effort was foiled by Subasic, parity was restored when Akinfeev got down to his left to keep out Mateo Kovacic's second attempt for Croatia.
For Croatia, Mateo Kovacic had his penalty saved, but Marcelo Brozovic, Modric and Vida all netted theirs before Rakitic stroked in the victor. A miss from Mario Fernandes and Luka Modric's goal gave an advantage to Croatia.
Smolov began the shootout by sending a soft shot that Subasic was able to paw down after diving to his right, and Brozovic put Croatia ahead.