The Bucks tasted defeat for only the 12th time this season with late fouls proving pivotal with the Bucks up at 116-115 with just over 20 seconds to play.
James Harden, who scored 24 on the night, added one in the last second for the victory, while Houston's top scorer was Westbrook on 31 points and eight assists.
The Nets sit eighth in the Eastern Conference, while defeat for Washington – their second on the bounce – means they stay ninth and need to make up three games with six to play to remain in the play-off hunt.
Joe Harris scored 27 for Brooklyn, while Caris LeVert proved pivotal in the dying minutes of the game, scoring a lay-up with 5:08 on the clock to break a 100-100 tie and adding 12 more before the final buzzer.
The battle for the Western Conference spots is still open after the eighth-placed Memphis Grizzlies were downed by the ninth-placed San Antonio Spurs 108-106, as DeMar DeRozan hit two free throws with a second left.
Elsewhere, Jayson Tatum scored 34 points as the Boston Celtics saw off the Portland Trail Blazers 128-124, the Orlando Magic swept past the Sacramento Kings 132-116 and the Phoenix Suns toiled to a 117-115 win over the Dallas Mavericks.