The victor of the match will face England in the semi-finals.
The quarterfinal at Sochi was tied 1-1 after regulation and 2-2 after extra time, forcing both Croatia and host Russian Federation to a shootout for the second consecutive game. Harry Maguire and Dele Alli were on target for England with two headers.
And he did - sending Akinfeev the wrong way to send the Croats into their first World Cup semi-final since 1998.
Kramaric had an overhead kick gathered by Akinfeev shortly after the break and Perisic was then inches away from putting Croatia in front as he saw his strike come back off the inside of a post.
But Mario Fernandes found Russia's equaliser in the 115th minute to send the match to penalty shootout, in which Croatia prevailed to set-up a semi-final clash with England, who beat Sweden 2-0 earlier in the day in Samara.
Four more boys safely out as day two mission ends
The 12 boys, part of a youth soccer team known as the Wild Boars, first went missing with their coach more than two weeks ago. The rescued boys were being treated by medics at the field hospital, and were later airlifted to a larger nearby hospital.
"We feel like conscripts, when they come to just before they have done their two-year (national) service, like those a bit demoralised who wanted to stay in the army for some more time longer".
Tied after 90 minutes, the game went to extra-time, and Croatia took a big step towards sealing victory after defender Domagoj Vida headed his team 2-1 up from a corner kick.
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.
Cheryshev was Russia's leading scorer at the World Cup with four goals.
Serena Williams loses first set, but reaches Wimbledon semifinals
Serena Williams , who lost a set for the first time at this year's Wimbledon , came back to win the final two sets 6-3, 6-4. Bertens will play 13th seed Julia Goerges for a place in the semi-finals after the German beat Croatia's Donna Vekic .
Russian Federation slowly grew with the game and took the lead out of nowhere, thanks to Cheryshev, who kocked down an aerial ball before playing one-two with Artem Dzyuba down the inside-left channel.
Fernandes sent the ball wide of the mark while captain Modric handed Croatia a 2-1 lead. And they had also scored in each of their last eight World Cup Games, a run that stretched all the way back to 2006 and a goalless draw with Japan.
"We should have finished the job before penalties, but maybe it's written in the stars that we have to go through the extra drama", said inspirational playmaker Luka Modric.
Croatia's Andrej Kramaric, right, scores his side's first goal during the quarterfinal match between Russia and Croatia at the 2018 soccer World Cup at the Fisht Stadium in Sochi, Russia, Saturday, July 7, 2018.
A look at what’s ahead in the World Cup
And the 19-year-old was overwhelmed at the thought of helping France reach the final in his first appearance at a World Cup . The players gave everything and were pushing until the last second to get back in the game. "We don't want to stop".