Dzyuba forced a penalty and scored from the spot to keep Russian Federation in the game after falling behind 1-0 Sunday, then roared on his teammates as they kept the 120-minute match deadlocked and won the penalty shootout 4-3 against heavily favored Spain.
"I don't believe I'm man of the match", said Akinfeev, a Moscow native, one of the evening's many heroes and his country's captain.
Spain dominated the possession during the game - completing more than 1,000 passes - but was too passive against a well-organized Russian defense.
Earlier Sergei Ignashevich struck an own goal before Artem Dzyuba equalized by converting a 41st-minute penalty.
Russian Federation will now play the victor of the match between Croatia and Denmark in the quarterfinals in Sochi on July 7 while Spain are eliminated.
"We were hoping for penalties", Akinfeev said.
"We are very relaxed and ready to do a good job", Cheryshev said Saturday, switching between fluent Spanish and Russian at a news conference.
Spain's Sergio Ramos celebrates his team's equaliser in Moscow (AP).
And it was Iniesta who nearly came up with the victor in the 85th minute, Akinfeev getting down low to his right to save, the goalkeeper then denying Iago Aspas on the rebound.
At the other end, concerns remain about their backline despite the presence of two of the best central defenders in Sergio Ramos and Gerard Pique.
He has not been the only player to impress for the host nation so far.
Tensions high as the rain fell, a flashpoint came late in extra-time when Ignashevich grappled with Pique in the box, but appeals were waved away after a VAR review and spot-kicks would be decisive.
Extra-time did little to create a sense of urgency for either team, although Spain did enjoy the best of the chances.
Spain won a first World Cup in 2010, before then they had never got beyond the quarter-finals.
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.
Dzyuba himself stepped up and sent David De Gea the wrong way.
The king heads FIFA's list of VIP guests for the round of 16 game, which also includes Russia's Prime Minister Dmitry Medvedev.
Mark Geiger has been assigned to referee Tuesday's World Cup round of 16 match between England and Colombia, four years after he became the first American to officiate in the knockout stage of soccer's showcase.
"Everything is changing, " Spain coach Fernando Hierro said through a translator.
Andres Iniesta's introduction also altered the pattern of play, with Spain looking more confident after his entrance.
Spain stopped Russia's European Championship run in 2008 - the country's best tournament performance since the end of the Soviet Union - and also played to a 3-3 friendly draw a year ago, though one in which Spain didn't field a full-strength team. But Denmark, unbeaten in 18 games coming into the game, provided tough opposition.