A 3 by Gary Clark tied the record with 2:42 left, and fans began chanting, "Three, three, three," every time Houston got the ball after that. The Rockets missed their next four tries before Michael Carter-Williams hit one with 31.1 seconds left and was greeted with loud cheers. The previous record was held by the Cleveland Cavaliers, who made 25 against Atlanta on March 3, 2017.
Harden made six 3s, Chris Paul added five, and Gerald Green and Eric Gordon made four each on a night when Houston attempted 55.
The win streak ties the Rockets' longest of the season and comes after they lost their previous three games.
Houston had a comfortable lead throughout most of this one, and seven straight points, capped by a dunk from Clint Capela, made it 111-94 with about nine minutes left.
Tomas Satoransky and Bradley Beal made back-to-back 3-pointers for Washington after that, but Harden ended the run with a 3 that left Houston ahead 114-100.
The Rockets then used a 12-3 run, with all their points coming on 3s, to make it 123-103 with about five minutes left. Harden and Gordon both made one in that span, and Green capped the spurt with two. Coach Mike D'Antoni cleared the bench soon after.
Beal had 28 points to lead the Wizards, who lost for the sixth time in seven games.
Houston led by 12 at halftime, and a basket by Paul left the Rockets up 80-64 with about nine minutes left in the third quarter. The Wizards used a 6-3 run, with four points from Wall, after that to cut the lead to 83-70.
The Rockets made four quick points later in the quarter, highlighted by a reverse layup by Gordon, to leave them up 91-76 with about three-and-a-half minutes left in the quarter.
Washington got within 11 late in the third quarter, but Harden scored the next five points, capped by a 3-pointer, to make it 101-85.
The Wizards scored the last five points of the quarter, with three free throws, to cut the lead to 101-90 entering the fourth.
The victory came after the Wizards won the first meeting against Houston this season 135-131 in overtime on Nov. 26, despite a season-high 54 points from Harden.
The Wizards often double-teamed Harden in the first half, leaving others open, which led to easy baskets in the lane and a 70-58 lead by the Rockets at halftime.
Wizards: Made 10 of 30 3-pointers. ... John Wall added 18 points with 12 assists. ... Trevor Ariza had 15 points in his second game since a trade from Phoenix.
Rockets: Gordon added 18 points. ... James Ennis missed his fourth straight game with a strained right hamstring. ... Houston had 15 turnovers.
Wizards: Host Phoenix on Saturday night.
Rockets: Visit Miami on Thursday night.
Rockets drain NBA record 26 3-pointers
The Rockets connect on an NBA record 26 3-pointers, James Harden hitting 6 of them and Chris Paul 5.