Kirileno told teammate Vitaliy Fridzon to take the winning shot in a group-stage victory over Brazil, and did it again before Alexey Shved buried the three-pointer that put Russia ahead for good.
"We are 2-for-2," Kirilenko said.
Kirilenko had limped off the floor of a semifinal loss to Spain on Friday with what coach David Blatt said was a quadriceps injury.
But he bounced back with a strong game against Manu Ginobili and the Argentines to earn a medal in his third Olympics and last game before returning to the NBA with the Minnesota Timberwolves.
"It's definitely a huge, huge deal," Kirilenko said. "If I would be 23 years old right now, I would be happy to get that medal for my collection. But right now I' more happy with the team priorities. ... Nobody can say we suck. We play tough basketball, we have good composure and pace of the basketball and other teams don't like to play us."
The bronze gave Russia 79 medals in London, behind only the United States and China, though their 21 golds ranked fourth, behind the U.S., China and Great Britain.