James finished with 43 points to extend his streak and break a tie with legendary Wilt Chamberlain. Hall of Famer Chamberlain also had the two longest streaks in history, runs of 65 and 31.
"He has great hands and great anticipation," said Houston coach Mike D'Antoni. "We ask him to do a lot and he just buckled down and did it tonight.'"
Harden added 12 rebounds, six steals, five assists and four blocks. He was 12 of 22 from the field, making four of 12 three-pointers, and made all 15 of his free throws.
Gerald Green added 25 points, and Kenneth Faried had 16 points and 12 rebounds as the Rockets snapped a two-game losing streak. They managed to hold the Jazz to 36 percent shooting and forced 23 turnovers.
"We smelled a little blood and went after them," D'Antoni said.
Donovan Mitchell finished with 26 points and nine assists and Rudy Gobert added 10 points and 13 rebounds in the loss.
Elsewhere, Dallas rookie Luka Doncic matched his season high 35 points to lead the Mavericks to a 111-98 victory over the Cavaliers in Cleveland.
Slovenian sensation Doncic, who won't turn 20 until February 28, had 28 points by halftime -- almost as many as the 30 points scored before the break by all of Cleveland's starters.
"I just felt great," said Doncic, who was cheered by a number of Slovenian flag-waving fans in Cleveland.
"It wasn't my best second half for sure, but we got a win so that's all that matters for me."
Doncic had missed Thursday's game in Detroit with a sore left ankle but showed no sign of injury as he scored 18 points in the first quarter -- when he became the seventh player to score 1,000 points in the NBA before turning 20, following in the footsteps of such stars as LeBron James, Kevin Durant and Kobe Bryant.
Doncic also pulled down 11 rebounds and handed out six assists.
Harrison Barnes scored 13 of his 17 points in the fourth quarter -- his three-pointer with 4:34 to play putting the Mavs up 15 points as Dallas ended Cleveland's two-game winning streak.
In Oakland, Klay Thompson returned from a one-game absence to score 28 points and Stephen Curry overcame a slow start to lead the Golden State Warriors to a 115-101 win over the Los Angeles Lakers, who were again missing LeBron James.
Curry missed his first eight shots but still finished with 14 points.
James' rest was for what coach Luke Walton described as "load management."
- James still sore -
The four-time MVP had been set for his second game back after missing 17 with an injured groin, but woke up Friday feeling "very sore" after playing 40 minutes in the 123-120 overtime win against the Los Angeles Clippers.
In Detroit, Lou Williams came off the bench to score 18 of his game-high 39 points in the fourth quarter for the Los Angeles Clippers, who erased a 25-point deficit to beat the Pistons 111-101.
Home fans were raining boos on the Pistons by the end of the night.
Blake Griffin led the Pistons with 24 points against his former team, but connected on just five of 17 shots over the last three quarters.
He made just three of 12 from three-point range.
The Clippers reserves out-scored the Pistons bench 80-17 as the Clippers posted the biggest comeback in team history.
"We just stayed the course," Williams said. "We didn't play our best basketball in the first half. We played our style the second half and gave ourselves an opportunity to win the game."
Milwaukee's Most Valuable Player candidate Giannis Antetokounmpo made all 17 of his free throws on the way to 37 points in the Bucks' 131-115 victory over the Washington Wizards.
Antetokounmpo added 10 rebounds for the Bucks, who notched a third straight win and remained two games ahead of Toronto atop the Eastern Conference.
Bradley Beal had 24 points for the Wizards, who are 9-7 since all-star point guard John Wall suffered a season-ending heel injury.