"Over the whole second half and extra time we were defending our goal and managed it, we were hoping for penalties because Spain are hard to beat".
Akinfeev, who started his professional career by saving a spot kick in his first ever match, then kicked Aspas' penalty high and wide to trigger wild celebrations. Dzyuba hit the spot-kick hard to David de Gea's right as the Spanish goalkeeper dived the wrong way.
Russian Federation and Spain finished extra time with the score at 1-1.
Asked why he did not appear to be as happy as his players, the 54-year-old said: "My emotions are simple - you show them when you direct your team but when it's over I'm already thinking about the next game".
"We are having a fantastic World Cup", said Akinfeev.
They will face either Croatia or Denmark in the quarter-finals and have every chance of winning if they can put on a defensive performance equal to today's effort and stifle their opponents.
Spain has gone three straight major tournaments without a title.
Spain, buoyed by their breakthrough, were beginning to dominate possession with Russian Federation struggling to find a foothold, but the hosts were briefly boosted when an Dzyuba header from a long free kick caused confusion before Spain whacked it clear.
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There are good players and it's important we return to the way which gave us so much success". "I am very grateful for one of the greatest players in our history".
But after consulting VAR the referee ruled out a penalty.
It was Nacho, in for Dani Carvajal at right-back, who won the free-kick that led to Spain taking the lead as he was caught by Yury Zhirkov to the right of the penalty area.
Akinfeev had to deny both Pique and Rodrigo late on, but his real heroics came in the shootout as he saved spotkicks from Koke and Iago Aspas to send Russian Federation into the quarter finals.
Akinfeev came to the rescue once again to deny Rodrigo after his fine sprint before becoming a national hero by saving the two spot kicks to send the hosts into the last eight and his fellow Russians to the streets to celebrate.
At this point, Spain had not had a shot on goal but twice in added-on time, Diego Costa attempted to regain his side's advantage, only to see a shot from an acute angle and a header saved by Igor Akinfeev.
A low 20-yard drive heading towards was palmed away by at Akinfeev at full stretch, with the angled follow-up from Aspas also pushed to safety.