They took the lead on 12 minutes through an own goal from Sergei Ignashevich, but Russian Federation equalized from an Artem Dzyuba penalty and kept the Spaniards at bay for the remainder of the game and extra time.
Dropped to bench for a tournament game for the first time in 12 years, Iniesta still almost won the game with Spain's best effort to break down the mass ranks of Russian Federation resistence.
"I had to persuade them this was the only way as they don't like this system", said Cherchesov, whose side had one shot on target in two hours of football.
Germany, Argentina and now Spain - seven World Cups between them - are out and there is a chance for someone new to contest the final on July 15. Spain, on the other hand, finished unbeaten in the during group-stage.
After the appeals died down and Pique was deservedly booked, Dzyuba sent de Gea the wrong way and his countrymen - who filled more than three-quarters of the stadium - wild with delirium. Regularly hit long balls straight back to opposition or out of play. Iniesta also scored the first spot-kick of the shootout. He was challenging Sergio Ramos when the free kick was floated in.
A defensive error let Russian Federation level in the 41st, after Gerard Pique's raised arm blocked a header by Artyom Dzyuba.
Sergio Ramos claimed the goal but replays showed that Ignashevich got the final touch with his heel as he battled the Real Madrid defender inside the box.
Cherchesov said his players would need a thorough rest before he could be sure of who he can choose from for the quarterfinal in Sochi on Saturday. Calmly converted in the shootout.
Igor Akinfeev assumed instant Yuri Gagarin status by saving two of Spain's spot kicks on Sunday. Safe and sound, but not penetrative, in possession.
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Modric didn't hide in the penalty shootout.
MF David Silva, 4 - Surprisingly kept his place in the XI, but again looked off the pace, losing the ball three times in the opening eight minutes.
David de Gea - 5 - Had little to do between the sticks until being called into action during the dying embers of the first-half with his first task being to prevent Russian Federation scoring a penalty - he failed as Dzyuba unleashed a thunderbolt into the back of the net. However, star man James Rodriguez is struggling badly with injury - if he misses out, England will really fancy their chances.
"Andres Iniesta, his last game probably, for Barcelona, for Spain, now going to Japan".
Now they must find a way past title contenders Spain to prolong their run after reaching the knockout phase for the first time in the post-Soviet era.
Cavani, who lines up alongside Mbappe at Paris Saint-Germain, opened the scoring with a leaping header and curled a superb first-time shot past Rui Patricio to win the match after Pepe had equalised.
So often, he has rescued points for United by taking to the air and clawing away a goal-bound strike and he started the World Cup as Spain's undisputed No 1.
FW Rodrigo Moreno, 7 - The fourth sub in extra time provided a rare spark of imagination with a dummy nutmeg to set himself up for forcing Akinfeev's most hard save of the game.