Russian Federation captain and goalkeeper Igor Akinfeev was the hero as the hosts stunned Spain 4-3 on penalties to reach the quarter-finals after a 1-1 draw in Moscow.
Spain controlled the game as expected but struggled to create genuine chances, with Akinfeev sharp on the few occasions they did before saving from Koke and Iago Aspas to seal a 4-3 win on penalties.
"The football god always seemed to be turning away from us, and I think today it's like never before", Dzyuba said.
A deep corner was met by Dzyuba, whose header struck the outstretched arm of Pique, with referee Bjorn Kuipers contemplating his decision before pointing to the spot.
Russian Federation will meet either Croatia or Denmark for a place in the semi-finals, as Spain's rotten run against hosts nation at major tournaments continued.
The older Cheryshev said the current Spain team "doesn't have the same cohesion" as back then, so Russian Federation "has a huge chance to beat Spain". "Great news for us". Spain join former champions Germany and Argentina as the fancied teams eliminated from the tournament.
Russian Federation were without Igor Smolnikov due to suspension and brought in Brazil-born Mario Fernandes to replace him at right wing-back. Russia, in turn, the next stage will play the victor of Croatia - Denmark.
"It's painful, there's nothing else we can say", Spain captain Sergio Ramos said.
Instead, Spain seemed entirely incapable on increasing the pace as, minute by minute, home hopes grew from a long-shot dream, through a gradual realization that their highly-ranked rivals were not actually that unsafe, to eventual delirious celebration.
Spain substitute Andres Iniesta stepped up first and scored, to which Fyodor Smolov replied.
It is unclear if interim coach Fernando Hierro will continue in a job he did not seek before Julen Lopetegui was sent home.
Many teams would have perhaps crumbled under these circumstances, but Spain have instead already bettered their disappointing performance at the 2014 World Cup in Brazil which saw them knocked out of the group.
Spain's early dominance pays off There's a thought that Spain, aside from its opener against Morocco, has been scoring some very un-Spanish goals at the 2018 World Cup.
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.
World Cup history was made with the first use of a fourth substitute, which Federation Internationale de Football Association now allows teams in extra time.
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. 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.
Spartak won the tie 3-1, with all the goals coming in the second leg in Madrid.
Dzyuba converted from the spot and it proved to be a costly mistake from the Barcelona defender as the rest of the match turned into defense versus attack with Russia's defenders, ultimately, coming out on top.