Aguero scored on the rebound in the 86th minute to give Argentina a seven-game unbeaten streak despite a difficult World Cup qualifying campaign.
It was the first game in four years at Moscow's vast Luzhniki Stadium, which will stage the World Cup final next year, though some fans voiced concerns that security forces and transport links were overloaded by the capacity crowd.
Argentina struggled to break down Russia's defense but coach Jorge Sampaoli saw encouraging signs.
"We played in a straightforward, courageous, bold manner. We created resistance to the pressure that our opponent put us under," Sampaoli said through a translator.
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On a chilly afternoon in Moscow, Aguero and Angel Di Maria went close to scoring for Argentina in the first half, but were denied by good saves from Russia captain and goalkeeper Igor Akinfeev.
Russia came closest to scoring with two efforts from midfielder Denis Glushakov around the hour-mark, but one went narrowly wide and the other was hit straight at goalkeeper Sergio Romero.
Otherwise, the hosts posed little attacking threat.
"We were bad in the transition out of defense," said Russia coach Stanislav Cherchesov, who thought Argentina's goal should not have counted because of an offside in the buildup.
"Our opponents were stronger today but we need to learn how to play right and beat them," Cherchesov said. With a sellout crowd, security was tight at Luzhniki. Many fans complained on social media that security kept them in the stadium for up to half an hour after the game and that police nearby forced fans down packed streets.
There were complaints it took up to 90 minutes to reach nearby subway stations on crowded roads lined with fences and police. That compares to 10 to 15 minutes normally.
The grand 1950s facade stays, but the stands were torn out and rebuilt closer to the pitch in a football-first configuration that does away with the Olympic athletics track.
The pitch is natural grass, not artificial as before.
The steeper stands, now over two main tiers, mean better visibility for fans even as capacity increased from the pre- refit 78,000 to 81,000 for the World Cup.