The Titans were put in position to win following Jason McCourty's interception of a Ryan Fitzgerald pass at the Tennessee 48-yard line.
Chris Johnson gave the Titans (3-4) a first down in Buffalo territory with a 27-yard run and Hasselbeck, facing a 4th-and-9 play, hit Washington in the back of the end zone with the game-winner.
The big play handed Tennessee its first road victory of the season.
Fitzpatrick finished 27-of-35 through the air for 225 yards, three touchdowns and a pick-off for Buffalo (3-4).
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