Jamal Crawford poured in 23 points and doled out three assists for the Knicks, who have won two in a row. Nate Robinson gave 17 points and six assists, while Wilson Chandler ended with 17 points.
Caron Butler had a game-high 30 points and seven boards in a losing effort for the Wizards, who have lost each of their four games to start the season. Antawn Jamison had 24 points and 12 rebounds, and Nick Young had 16 points off the bench.
With the Wizards down by five late on in the fourth, Young banked in a jump shot, Jamison connected on a layup and Butler followed suit with a layup to give Washington its first lead since the second quarter, 103-102.
Chris Duhon, though, immediately drained a three-pointer to spark a 10-0 New York run. Crawford added four points on the burst, while Randolph's layup capped it with 2:30 left in play to give the Knickerbockers a 112-103 lead, which held up through the end of the game.
The Knicks held a slight 29-28 lead after the first quarter, and opened up a 64-58 lead at the half.
An Etan Thomas basket brought the Wizards within six points, but Crawford responded with a three, Randolph dropped in a jumper and Chandler connected on a bucket to give New York its biggest lead to that point, 76-63, midway through the third quarter.
However, Washington outscored its counterpart 18-6 over the next five minutes to make it a game again, 82-81. Jamison sparked the surge with four points and JaVale McGee closed it out with a three-point play.
The teams traded baskets and the visitors still led by one, 89-88, heading into the final frame.
David Lee's turnaround hook shot sparked an 8-2 Knicks run to begin the fourth. Robinson had four points and Anthony Roberson added a layup to give New York a 97-90 lead with 9:22 left in the contest.