Udrih put the Knicks back in front by making one of two free throws with 24 seconds left.
After a Wizards timeout, Beal ran down the clock before driving past Udrih and hitting the game-winning shot.
The curious decision by the Knicks followed.
"Well it's easy to stop playing, make a basket, anticipate a timeout, start walking to the bench, and now all of a sudden you got one of the best players in the league come at you full bore," Wizards coach Randy Wittman said. "Even after the ball went in, we stayed up. I think we had two guys flying at Carmelo at the end."
The Wizards led 53-42 at halftime and increased their lead to 59-44 on Ariza's short jumper with 9:12 left in the third quarter.
The Knicks then went on a 12-0 run, highlighted by consecutive 3-pointers by J.R. Smith and capped by Anthony's layup. New York went ahead 66-64 on Anthony's jumper with 3:06 left in the quarter and increased it to a game-high at 77-71 on another jumper from Anthony with 33 seconds left before settling for a 77-75 lead after three quarters.
After shooting just 15-47 (31.9 percent) in the first half, the Knicks went 13-19 (68.4 percent) in the third quarter.
Smith finished with 18 points. Tim Hardaway Jr. scored 14, Bargnani had 13 and Udrih with 12.
NOTES: Pablo Prigioni was taken out of the game with 9:49 left in the third quarter with a hairline fracture of his right toe. He is expected to be out at least two weeks. He was replaced by Udrih. . Smith returned to the starting lineup after sitting the entire fourth quarter in Saturday night's Knicks win over Atlanta. . Wittman and Woodson were teammates at Indiana University in 1979-80.