The Blackhawks were trailing 4-3 with 12:05 remaining in the game when Brandon Saad converted a pass at the net from Michal Handzus for the equalizer.
Then with 6:07 left, Oduya launched a slapshot from the left circle that deflected off the stick of the Capitals' Joel Ward, went off goalie Braden Holtby's pad and into the net for the decisive tally.
Corey Crawford stopped 28 shots to record the win.
Mikhail Grabovsky supplied a three-goal hat trick and added an assist for the Caps.
Holtby turned aside 29-of-34 shots for Washington.
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