Russia's defence then held firm as Spain dominated possession before the heroics of Akinfeev triggered wild celebrations in the rain.
However, Russia got a penalty five minutes before the break with Gerard Piquè penalised for a raised arm.
Russian Federation will now meet Croatia in the Quarter Final at the Fisht Olympic Stadium, Sochi on Saturday, July 7, 2018.
Rodrigo carried Spain's biggest scoring threat and again forced a smart save from Akinfeev, but Russian Federation stood firm to force the match into a penalty shootout.
Missed penalties from Koke and Iago Aspas ensured Russian Federation progressed to the last eight for the first time in its history, sparking joyous celebrations at Moscow's Luzhniki Stadium.
"We're insanely happy", Dzyuba said.
It was the classic "tiki-taka" style that led to a World Cup title in 2010, bookended by a pair of European Championships.
Still, World Cup history was on Russia's side after its well-organized team survived two hours of barely having the ball.
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 previous year, though one in which Spain didn't field a full-strength team. Spain played as cautiously as any team at the World Cup. In the end, the 2010 World Cup winners were made to pay for their lack of ruthlessness. Sergei Ignashevich seemed more concerned by bulldozing Sergio Ramos down in the six-yard box than defending the Spain corner, and with his back to the ball accidentally clipped it over the line to put Russian Federation behind in the 11th minute.
In overtime, Spain has created a few more chances, but Akinfeev played correctly. As a result, the fate of the tie was decided in penalty shootout, where the bad beats Koke and ASPAs cost "red fury" places in the quarterfinals. During that game, injury-prone center half Vincent Kompany went on in the second half for his first game since injuring his groin on June 2. However, Cherchesov refused to get carried away and spoke like a man on a mission after his side booked a quarter-final clash with Croatia or Denmark.
Dzyuba himself stepped up and sent David De Gea the wrong way.
After Akinfeev acrobatically blocked the left-footed shot of Aspas, he leapt up, punched the air with both hands and dove into a belly flop cross the rain-soaked turf as teammates raced to him from the halfway line.