After goals from Alexander Golovin and Denis Cheryshev, Iago Aspas had to score to keep Spain alive, but saw his penalty saved by Akinfeev's outstretched left leg, sending the hosts through to the quarterfinals.
Spain dominated the first 30 minutes, pressing Russian Federation heavily as the hosts struggled to advance beyond the halfway line in humid conditions.
"It's the coach who makes the decisions", Iniesta said.
Pique, one of three starters left who played in the 2010 final victory over the Netherlands, was called for a blatant hand ball, leaping into a jumping jack as Artem Dzyuba's header bounced off his left arm.
Rodrigo, an extra-time substitute, had a decent sight of goal in the second half of stoppage time, beating his man one-on-one before drilling a shot Akinfeev did well to save.More news: Clippers DeAndre Jordan opts out of contract, becomes free agent
This match also witnessed history as Russian Federation became the first team in World Cups to make four substitutions.
Russian Federation continued to sit deep and hit Spain on the counter-attack once in a while, which eventually paid off when they equalised five minutes before half-time with the only shot on target they managed in 120 minutes of open play.
But for all the talk of tactics and generational change, Spain's 2018 World Cup will be remembered for a simple act of self-sabotage.
The World Cup's lowest-ranked team - 70th when the tournament began - now advances to a quarterfinal against Croatia or Denmark in Sochi on Saturday.
Fernando Hierro & Co were shocked by hosts Russian Federation in Moscow, losing 5-3 on penalties after drawing 1-1. "We might have attacked more if we only played four at the back or with more forwards but I believe we were victorious because my players stuck to the plan".
"Let us say this openly, they are better than us in many ways, " he said.More news: Germany coach Joachim Low ponders over future after defending champions' meek exit
The extra defensive duties offered by the two newcomers allowed Spain to control the game with its midfield purring on to control 75 percent of possession, but the team had only three shots on target to show for its trouble (in the first half). He must hope the message belatedly gets through in time for their encounter with the tournament's joint-second highest scorers.
"The reality is that this was my last match with the national team", the 34-year-old Iniesta said.
But this time, they just passed side-to-side, sometimes backwards, nearly never to attack.
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. While Russia arrived at the finals win-less in eight months before finding some badly-needed form, one of their better results in that period came in a 3-3 draw with Spain last November.
"We can't start finger-pointing at anyone and laying any blame here", he said.More news: LeBron James’ free agency: Reports from around the NBA
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