Spain, the 2010 world champions, controlled the game with their slick passing but did little more, failing to create another scoring chance in the first half.
Aspas almost picked out Carvajal with a cut-back seconds into the start of extra time, which saw Aleksandr Erokhin come on as Russia´s fourth sub - the first time the rule has been applied at the World Cup.
However, Russia can expect help from the 81,000 fans in the Luzhniki Stadium and will also take heart from the fact Spain have never beaten a host nation in a major tournament.
The 2018 FIFA World Cup kicked off on June 14 and will continue until July 15, with the matches being played in 11 cities across Russian Federation. Spain's first corner of the match followed after 59 minutes, but Isco's cross was headed away by Dzubya.
Spain could not find a way past a solid Russian back line after taking the lead through a Sergei Ignashevich own goal on 12 minutes from a free-kick, an error perhaps forced by Spain captain Sergio Ramos, who tussled with the defender in the area as the set piece was delivered. Subs - Granat 5, Smolov 6, Cheryshev 6, Erokhin 5.
Dzubya stepped up to blast nervelessly past David De Gea and level the match after 41 minutes, and there was still time for Isco to see a shot blocked and Diego Costa to force Akinfeev into action from an Isco cross.
Fernando Hierro's charges continued their dominance after 20 minutes but they could not get through the hosts' defence in open play.
This will go down as one of the biggest shocks of the tournament, with Spain failing to make their dominance of possession count as Russian Federation held firm and stuck to their defensive gameplan.
Russian Federation goalkeeper Igor Akinfeev's save on Iago Aspas' penalty kick took less than a second, but it still made for the ultimate swing of emotions for the Spanish forward.
Despite the added time of two minutes, both times retreated for the 15-minute break with the scoreboard flashing 1-1.
Everyone inside the Luzhniki Stadium, and all those watching at home, held their breath as the referee and the VAR team checked footage for holding on Gerard Pique and Sergio Ramos late on in extra time.
But it didn't matter, Denis Cheryshev scored Russia's fourth before Iago Aspas' spotkick was again saved by Akinfeev, sending Spain packing.