Veteran goalkeeper Akinfeev saved penalties from Koke and Iago Aspas as Russian Federation won the shootout 4-3, reaching the last eight for the first time since 1970, when they played as the Soviet Union.
There was more penalty drama in the later match in Nizhny Novgorod, with Croatia coming out on top 3-2 just minutes after Denmark goalkeeper Kasper Schmeichel had saved a penalty at the end of extra-time.
Dzyuba stayed alone up front, using his size and strength to disrupt Spain's defense, win headers and hold up the ball. "I told him, "We know the price you have paid to climb back to this level", said Tite.
Akinfeev saved a bad Koke penalty to put Russian Federation in control and with David de Gea unable to stop any of the four kicks he faced, Aspas' effort was brilliantly blocked by the goalkeeper to spark chaotic celebrations.
It was a shocking elimination for Spain, joining former champions Germany and Argentina.
Still, World Cup history was on Russia's side after its well-organized team survived two hours of barely having the ball. Why are we looking in the rear-view mirror when we're in the fourth game of the World Cup?
Akinfeev ensured that, in the 32-year-old captain's 111th game for his country. The Atletico Madrid midfielder lifted his shirt over his eyes in dismay. They had a rare chance to break the deadlock in the 85th minute when Aspas chested a Jordi Alba cross into the path of Iniesta, whose shot forced a lunging save from Akinfeev.
"We were hoping for penalties", Akinfeev said after the match ended 1-1 after extra time, following a Sergei Ignashevich own goal and an assured Artem Dzyuba penalty.
Russia, on the other hand, could only manage to keep the ball for 21 percent of the time, as the 2010 champions didn't allow them to settle.
It was too passive a performance by the Spanish, who were ranked 10th and among the pre-tournament favorites.
Spain will dominate the possession for large spells of the match, but it's obvious that under new management, Fernando Hierro, they aren't completely comfortably defensively, or when they're defending under fast transitions. "We left our soul in the pitch". Iniesta entered the game in the 67th-minute while converting a penalty shot in the shoot-out. He also converted his spot-kick in the shootout.
Russian Federation won the penalty for handball when Dzyuba's header hit Gerard Pique's arm at a corner.
In pretty much his only action before the shootout, goalkeeper David De Gea dived right as the Dzyuba's third goal of the tournament went to his left.
Spain were many people's favourites heading into this tournament, and they have shown in patches so far why that was the case. "We believed that order, discipline and dedication can beat class". "Nobody believed we could win and hopefully we can repeat that success tomorrow".
The match, which took place at the stadium "Luzhniki" and which was attended by thousands of spectators (among them the king of Spain), was hard and stressful.
Spain then took the lead from a more accurate set-piece in the 12th minute.