Denis Cheryshev gave Russian Federation the lead with a shot into the upper corner in the 31st minute and Croatia equalized with Andrej Kramaric's header near halftime.
The winners will face England in the semi-finals in Moscow on Wednesday.
Croatia scored their 10th goal at the 2018 World Cup, their second-best yield for a single tournament, one less than what their third-placed 1998 team managed on debut in France. "But he said we should have our eyes open and make the next steps". "We are together as one and anything is possible", Rakitic said.More news: HPV Test More Accurate than Pap In Screening for Cervical Cancer
Both goalkeepers made early saves in the shootout, with an injured Danijel Subasic stopping the opening shot from Fyodor Smolov.
FIFA has fined the Croatian soccer federation over $70,000 for violating media and marketing rules at the 2018 World Cup in Russia. "Croatia is a better team, but it would not be the first time the host, who before the championship was an outsider, go as far as, let's say, South Korea in 2002", says The Washington Post.
This is the first time in 16 years that two teams took a quarterfinal to extra time, after having played extra time in the round of 16 as well. "We gave everything we could".
Into extra time for the second consecutive match, Croatia looked exhausted with Mandzukic and Subasic limping, yet Sime Vrsaljko had to come off injured to use up their fourth substitution.More news: Israel Tells Khan Al-Ahmar Palestinians Leave by Friday
Russian Federation reached the quarter-finals for the first time since not playing as part of the Soviet Union, and the team exceeded expectations in knocking out Spain in the last round before running a much-fancied Croatia team incredibly close in Sochi.
And it was Lovren's team that prevailed 4-3 in the shootout when Barcelona star Ivan Rakitic netted the decisive kick.
Cheryshev was Russia's leading scorer at the World Cup with four goals.More news: Hurricane Beryl Is Tiny, Unpredictable