Spain followed suit as Rodrigo replaced Asensio, and the former went closest to ending the impasse, sprinting towards goal after a clever dummy, only for Akinfeev to claw his shot away before the rebound from Koke was blocked.
Sergei Ignashevich - 5 - Sergio Ramos got the better of the veteran defender with 12 minutes on the clock, as the 38-year-old wrestled the Real skipper to the ground before his back leg deflected the ball into his own net.
Spain was more urgent in extra time after being too passive when it dominated the ball in the 90 minutes of regular time.
Aspas almost picked out Carvajal with a cut-back seconds into the start of extra time, which saw Aleksandr Erokhin come on as Russia's fourth sub - the first time the rule has been applied at the World Cup.
The 34-year-old, who scored the winning goal at the 2010 World Cup final, won 133 caps in an global career spanning 12 years, during which he also won European championship titles in 2008 and 2012.
Despite 75 percent possession, Spain's famed tiki-taka passing game rarely carried them into the box and they had to wait until the 85 minute to test Akinfeev with a low drive from substitute Andres Iniesta. The loss continues their streak of never having beaten a host nation at the World Cup.
For the second time Russian Federation advance to the World Cup quarter-finals.
"In 2008, 2010 and 2012 we had the players we had and we played in a style no-one had before", Hierro said when asked about those great tournament-winning Spain teams and what had changed since then.
Novak Djokovic speaks out after reaching Queen's semi-finals
I was just thinking how I can return his serve and how I can get into play and position myself well in the next point. There's a lot of grass court players. "Last year I had a match point and couldn't convert it, this year it was Novak.
Artem Dzyuba's spot-kick gave Russian Federation something to hang onto, and hang on they did.
"Let us say this openly, they are better than us in many ways, " he said.
Isco produced a series of regal touches with Spain's King Felipe VI among those in the crowd at the Luzhniki Stadium, but Russian Federation equalised shortly before half-time.
Andres Iniesta of Spain consoles team-mate Jordi Alba following their defeat to Russian Federation.
Spain joined fellow heavyweights Germany and Argentina in making a premature exit, raising questions as to the wisdom of sacking Julen Lopetegui on the eve of the competition. It is a reality that is my last match with the national team - at an individual level a wonderful stage is over.
Iniesta, who struggled physically in recent years, converted Spain's first penalty in the shootout with what was his last ever shot at a World Cup.
The Spanish stopper is clearly not accustomed to criticism since recovering admirably from a shaky start to life at United but supporters tore into his distribution during Spain's final World Cup game.
Spain starting XI: De Gea; Nacho, Pique, Ramos, Alba, Koke, Busquets, Silva, Isco, Asensio, Costa. "But when you are sure about the way you have done things with responsibility and conviction, the only way is to move forward".