Alexander Sherov also scored for Russia, which advanced to a semifinal in Toronto against the winner of the Sweden-Finland quarter-final. Goalie Igor Shestyorkin was solid as the Americans outshot Russia 41-25.
Defencemen Anthony DeAngelo and Zach Werenski scored for the Americans, who will finish out of the medals for a second straight year after winning the 2013 tournament in Ufa, Russia.
The game that was neither chippy nor very physical was marked by a total of 14 minor penalty calls, helped by some embellishment from both teams.
The U.S. was drawn into five minor penalties in the opening period and fell behind 2-0.
After Tyler Motte and J.T. Collins were sent off, Barbashev poked in a loose puck in the crease on the two-man advantage at 2:31.
Sherov found another loose puck in front of Thatcher Demko to push in at 15:25 after Maxim Mamin's pass went off Brandon Carlo's skate in the crease.
The Americans got their first power play 2:02 into the second frame, but it took until 12:43 to score their first goal. They scored on a two-man advantage when DeAngelo wired a point shot to the top corner.
Russia restored its two-goal lead when Tolchinski's deflected shot got past Demko 1:27 into the third.
The Americans then failed to score during a second two-man advantage. They cut into the lead at even strength at 8:56 when Werenski threw a puck from the point that got through traffic and past goalie Shestyorkin.
A sparse crowd turned out at the Bell Centre for the weekday matinee, with pockets of Russia fans providing most of the chants and noise.