Hosts Russia has been the surprise package of the tournament and endured a real battle against Spain eventually prevailing after penalties.
Danijel Subasic saved another penalty and Ivan Rakitic converted the decisive one as Croatia ended the fairy-tale run of World Cup hosts Russian Federation with a 4-3 win on penalties on Saturday to set up a semi-final date with England.
The teams then went back and forth in regulation, and neither could find the victor, sending the game into extra time.
Denis Cheryshev's swerving left-footer from the edge of the box had opened the scoring for Russian Federation in the 31st minute of normal time but an unmarked Andrej Kramaric headed in a cross from Mario Mandzukic eight minutes later to restore parity. This media house does not correct any spelling or grammatical error within press releases and commentaries.
"We should have finished the job before penalties but maybe it's written in the stars we have to go through the extra drama", said Luka Modric, whose penalty bounced off Akinfeev's hand and the post before entering the other side of the net.
The crowd erupted, but Croatia were undeterred and equalised just eight minutes later.
Ivan Perisic came closest to scoring in the second half, while Domagoj Vida put the Blazers ahead in the first half of extra time, only for Mario Fernandes to restore parity at the death and force penalties.
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Cheryshev collected the ball near the halfway line, exchanged passes with Artem Dzyuba and curled a 25-metre shot past Subasic. 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.
Croatia will now hope to go one better than 20 years ago when they meet England on Wednesday at Moscow's Luzhniki Stadium, for a place in the July 15 final in the same arena.
An injury to Sime Vrsaljko forced Dalic to send on Vedran Corluka as his fourth substitution, though Croatia were celebrating when Vida nodded past Akinfeev.
But Russia themselves also had just one shot on target in two hours of football and needed two saves by their goalkeeper Igor Akinfeev in the penalty shootout to advance to the last eight and send the country into a soccer frenzy. Croatia equalized with Andrej Kramaric's header near halftime.
Croatia started the second half more urgent than they did the first and looked the more unsafe out of the two sides.
- There were three headed goals in this match - the most in a World Cup match since Germany 8-0 Saudi Arabia in 2002 (five).
With neither side able to find a victor the match headed into extra-time for the second time in a row for both teams.