Fedor Smolov, making his Russian national team debut, gave the hosts the lead in the ninth minute following a giveaway by Danny Williams. Bradley tied it on a 19-yard volley, his 11th international goal.
Roman Shirokov put Russia back ahead 2-1 when he converted a penalty kick in the 84th minute after a foul by Clarence Goodson, who pushed over Artem Dzyuba in front of goalkeeper Tim Howard following a free kick.
The 27th-ranked United States, missing regulars Clint Dempsey, Landon Donovan and Steve Cherundolo, are preparing for the final round of qualifying next year for the 2014 World Cup. They finished the year at 9-2-3 with a .750 winning percentage that tied their best, set in 2005.
The ninth-ranked Russians won their fifth straight match, improving to 5-0-1 since former England coach Fabio Capello took over last summer.
Jozy Altidore started for the U.S. after he was dropped from the roster for last month's World Cup qualifiers following poor performances against Jamaica in September, and Timmy Chandler started at right back. Chandler had not played for the Americans since last November, uncertain whether he wanted to commit to the U.S. program.
Russia went ahead early after the U.S. was given a free kick in its own half. Williams mis-hit the ball as Altidore crossed in front of him. Alexander Kerzhakov started a quick break and passed to Smolov, whose right-footed shot from 15 yards went under Howard's left arm. Smolov fell to the ground, injured, and was replaced by Maxim Gregoriev two minutes later.
U.S. captain Carlos Bocanegra left in the 18th minute with what the team said was an injured hamstring. He was replaced by Goodson, who paired with Geoff Cameron in central defense.
Howard made a fine double save in the 66th, batting away an open shot by Renat Yanbaev, then parrying Vasily Berezutskiy's shot over the crossbar.
Four minutes later, Fabian Johnson made a fine run up the middle of the field and passed to an open Altidore, who failed to control the ball.
Bradley scored after Maurice Edu passed to Juan Agudelo in the penalty area, and Agudelo headed the ball back to an open Bradley for a right-footed shot that clanked in off a post.