Washington, who finished with four passes for 92 yards, gave Tennessee a 23-16 lead with 4 1/2 minutes left.
But the Titans (9-7) only recorded the victory after Houston botched a potential game-winning, 2-point conversion with time running out.
Backup quarterback Jack Delhomme brought the Texans to within a point with 14 seconds left when he found Bryant Johnson in the back of the end zone for a five-yard TD.
AFC South-champion Houston (10-6) decided to go for the win with a 2-point conversion, but tight end Joel Dreessen drew a false-start penalty on the first try and then the snap sailed over Delhomme's head on the second attempt.
Tennessee recovered the loose ball and sealed the victory, keeping alive its slim playoff hopes.