The decisive penalty once again fell to Rakitic, though, and he confidently rolled the ball into the bottom-left corner to keep Croatia's dreams of a first World Cup final alive.
Ivan Rakitic scored the deciding penalty after a 2-2 draw.
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.
The Croatians will face England in the last four on Wednesday after the English booked their spot with a 2-0 win over Sweden in their quarter-final contest earlier in the day.
Gareth Southgate's young squad became just the third England team to make the semi-finals as they continued to make a mockery of dampened pre-tournament expectations with another impressive display.
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.
Denis Cheryshev opened the scoring in the 31st minute, drilling a left-footed shot from outside the box between two defenders and into the upper left corner for his fourth goal of the World Cup, bringing the crowd in Sochi's Olympic stadium - and all over Moscow - to full roar.
Though the hosts grew in confidence, Croatia took the lead in the 11th minute of extra-time as Domagoj Vida rose highest from a corner to head past Igor Akinfeev.
It looked as though another upset could be on the cards when Denis Cheryshev's stunning strike gave Russian Federation a first-half lead.
Don't miss out on the latest news and information.
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.
Russia and Croatia needed well over 90 minutes to decide a World Cup quarterfinal, with the Croatians winning in penalty kicks Saturday in Sochi, Russia.
Slack defending allowed Croatia to level eight minutes later.
"We have two matches to play, we are very motivated, we will give our all".
But with no substitutions remaining, Subasic had no choice but to continue, and he got off to a good start, saving the first penalty kick from Russian Federation striker Fedor Smolov.
Cheryshev was Russia's leading scorer at the World Cup with four goals. A Vedran Corluka goal put Croatia in front in the first half of extra time before Mario Fernandes levelled in the second.
Subasic was then tested again in the second period of extra time when he was able to corral Daler Kuzyaev's half-volley from the edge of the box, then stretched to cut out a cross. Just before the hour mark Akinfeev was again called into action but this time the man who could do no wrong for Russian Federation hesitated when collecting a cross.