In the regular time, Russian Federation broke the ice first in the 31st minute when Villarreal midfielder Denis Cheryshev completed a neat one-two with forward Artem Dzyuba at the edge of the box, opened up space with a touch and swerved a stunning strike into the top left corner.
Croatia, having gone close in the 60th minute when Ivan Perisic hit a post, then had the lead in the 11th minute of extra-time as Domagoj Vida nodded in, before Fernandes brought things level late on once again with a header of his own.
Russian Federation had just lost the opportunity to reach the semifinals in the World Cup they are hosting.
After ending Russia's impressive 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.
Ivan Rakitic had a good opportunity from a free kick just outside the box after a quarter-hour but could not keep his shot near the target.
Cheryshev collected the ball near the halfway line, exchanged passes with Artem Dzyuba and curled a superb 25-metre shot past the transfixed Subasic. Andrej Kramaric tied the score at 1-1 in the 39th minute.
The Blazers had the first sniff at goal after seven minutes when Ante Rebic headed over after meeting Dejan Lovren's header back into the danger area.
Yuri Gazinskiy became Russia's third substitutes, coming in for Dzyuba, but Croatia almost ran into trouble after making two more subs, Josip Pivaric and Mateo Kovacic on for Ivan Strinic and Kramaric. 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.
"After we went 2-1 ahead, we should have maintained our focus, but there was a lapse of concentration". His hand was forced when Sime Vrsaljko went down with a knee injury early in extra time and Dalic was forced to use his final substitution to replace the defender.
Akinfeev was helpless as Vida's header sneaked through a body of players into the bottom-right corner of the net from a corner on the right.
Croatia goalkeeper Danijel Subasic once again came up big, making some big saves in regulation, one after getting injured, and saving Russia's first penalty kick. Akinfeev denied Mateo Kovacic and nearly kept out Luka Modric's effort from 12 yards out, but Rakitic converted the decisive effort to seal a 4-3 win in the shootout.
The hosts have surprised everyone at this World Cup and will look to continue that.
The subsequent shoot-out then saw Subasic and Akinfeev make a save each before Fernandes' miss, which allowed Rakitic to send Croatia through.