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3 year oldWestbrook reached double digits in rebounds, points and assists for the 182nd time in his career.
The 32-year-old finished the night with 28 points, 13 rebounds and 21 assists.
However, Westbrook missed a possible game-winning buzzer-beater as the Wizards lost 125-124.
"I'm truly grateful for moments like this," he said of his record-breaking achievement. "Normally, I don't like to pat myself on the back, but tonight I will."
Robertson, 82, who played from 1960 to 1974, said he was "very happy" for Westbrook.
"Your family is very proud of you, and I salute you for all your accomplishments in the game of basketball," he added.
Magic Johnson - who sits third on the triple-double list - said he was "proud" of Westbrook and fifth-placed LeBron James called it "an incredible feat".
Elsewhere, Stephen Curry scored 36 points as the Golden State Warriors beat Western Conference leaders the Utah Jazz 119-116.
The Memphis Grizzlies secured a place in the Western Conference play-in tournament with a 115-110 victory against the New Orleans Pelicans, while the San Antonio Spurs moved a step closer to that event with a 146-125 win against the Milwaukee Bucks.
The Portland Trail Blazers beat the Houston Rockets 140-129 and the Indiana Pacers claimed a 111-102 victory at the Cleveland Cavaliers.
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