Zlatko Dalić's side has played some of the best football of the tournament and will be looking to emulate the heroes of 1998 who reached the semi-finals of the tournament in their first outing as an independent country.
Croatia will now face England in the last four for a place in the final.
A Vedran Corluka goal put Croatia in front in the first half of extra time before Mario Fernandes levelled in the second.
Both goalkeepers made early saves in the shootout, with an injured Danijel Subasic stopping the opening shot from Fedor Smolov.
At 1-1, Fernandes missed his shot - only the second player to miss in any of the four shootouts at this year's World Cup. 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.
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. Responding to Russian attack, Andrej Kramaric headed in an equaliser in the 39th minute of the game.
It was the second straight time both teams played in a shootout.
Rare chances for Stanislav Cherchesov's team saw Dzyuba head tamely at Danijel Subasic from a 62nd-minute corner, before substitute Aleksandr Erokhin sent his header well over the bar from Fernandes' cross on the right 10 minutes later.
Croatia also progressed to the quarter-finals thanks to a penalty shoot-out victory over Denmark.
"We looked composed", England captain Harry Kane said. "He congratulated us on a very good game".
"The president invited the head coach and team members to an event to sum up the results of the championship and discuss Russia's World Cup heritage", the Kremlin spokesman said.
Russia's Denis Cheryshev started the scoring in the second match of the day with a goal that is a shoo-in for any list of great World Cup knockout goals.
"I'm not sure anything went wrong as such", he said. "I think we primarily must focus on ourselves because I think our team has bigger quality, we just need to show that on the pitch and we will be able to set the tone", Rebic told reporters during the week.
It was predictable in many ways, therefore, that the deadlock was broken from a set piece.
There was another against Sweden, with Maguire heading in a driven corner from Ashley Young in the 30th minute.
The importance of the keeper's contribution was swiftly underlined with England doubling their lead in the 59th minute as Jesse Lingard's teasing cross into the box was met by Alli unmarked at the far post to head powerfully past Olsen. "But now we are collecting those debts in this World Cup". That helped them win a two-leg playoff against Italy to get to the World Cup, top a group containing Germany, and then beat Switzerland in the round of 16.