Russian Federation also were toothless up front until Gerard Pique's handball gave them a chance to level with a penalty in the 41st minute.
Spain has now failed to win a knockout game at three major tournaments since it won Euro 2012.
The home crowd greeted the arrival of extra time as if they had won the World Cup but Spain, seeing several of their rivals dropping to the floor with cramp and exhaustion, were no doubt confident that the hosts would eventually crack if they just kept rolling the ball from side to side.
"In the second half, we were trying to defend".
A stodgy Spain could not break the hosts down and were denied a penalty following a VAR review in extra-time, sending the tie to a shoot-out.
The Russians, who covered more ground per-player than any other side in the group stage, deserve enormous credit for the work they put in to stay tight and close down every threat.
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However, the two-time reigning NBA Finals MVP reportedly privately suggested the prospect was not as alluring as some might think. On Tuesday, Yahoo Sports NBA insider Shams Charania appeared on "The Herd" and suggested Leonard was now considering his options.
Akinfeev's foot kept Russia's party going Sunday night, ended the worldwide career of one of Spain's biggest stars and sealed the biggest upset of the tournament and one of the greatest in World Cup history.
Lamenting his side's failure to make more of their chances, he said it was a great shame two years of hard work and good results had come to an end "in a shoot-out which is basically a lottery and we weren't lucky". "We understand that in these moments you have to be effective".
Spain's chances to win the World Cup are now +450, while the Red Fury is -120 to reach the semi-finals.
The home team's only attacking tactic was launching long ball towards big front man Dzyuba, and that bore fruit when they were gifted an equaliser as Gerard Pique lost the flight of the ball from a corner and the striker headed it goalwards against the arm of the Barcelona man, who had inexplicably left it pointing straight in the air.
Dzyuba then smashed the ball home for the game's first shot on target and his third goal of the tournament.
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Some suggested that he might do the same to Cohen, long considered Trump's "fixer", but there is an apparent rift between the two. Of the celebrities that have verbally opposed Donald Trump as President, Sacha Baron Cohen is arguably among the loudest.
Spain finished the half with a burst of activity, registering a couple of efforts, but the next 40 minutes were turgid, a state of affairs not helped by the 25-degree temperature.
Spain, who have never beaten a host at World Cups or Euros, went ahead after 12 minutes when 38-year-old Sergei Ignashevich bundled into Ramos at the far post, knocking a free kick in with his heel for the 10th own goal of the tournament.
On the outside, it might seem like such a trivial thing but the importance of the crowd support can not be ignored - all the more so considering this is Russia's first time in the knockout stage of the tournament.
King Felipe VI of Spain was at the Luzhniki Stadium in Moscow.
Rodrigo almost won it five minutes later, when he beat his man with a dummy, sprinted into the box and fired a shot Akinfeev again did well to save. His second effort was one of just three of Croatia's six spot-kicks on the night to beat Kasper Schmeichel as the Dane's heroics couldn't prevent a 3-2 shootout defeat thanks to three stunning saves from Danijel Subasic. "We feel like all Spaniards this evening", he said. "To be a strong team is one thing, but you have to be the best on the pitch".
Despite 75 percent possession and more than 1,000 passes, Spain's famed tiki-taka game rarely carried them into the box and they had to wait until the 85th minute to test Akinfeev with a low drive from substitute Andres Iniesta.
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Minutes after fleeing the scene, the perpetrator attempted to carjack someone nearby, and fired more shots, according to the Star. Earlier, a police official had confirmed on Twitter that no students or school employees were involved in the shooting.