James has scored 20 or more points in each of the Heat's 24 games this season, which equals an NBA record, while the overall run, stretching back to last season and taking in post-season games, is at 45.
James also had nine rebounds and seven assists, while Marvin Williams led the Jazz with 16 points.
Kobe Bryant scored six of his 34 points in overtime as the Los Angeles Lakers stormed from behind to beat the Golden State Warriors 118-115.
The Lakers trailed by 14 at the start of the fourth, but they roared back to force the extra period and Bryant, who also grabbed 10 rebounds, brought them home.
Al Horford paced Atlanta with 20 points as all five Hawks starters hit double figures in a 92-75 win over the Chicago Bulls.
Horford also added 10 rebounds for his 13th double-double of the season.
Damian Lillard scored 25 points to help the Portland Trail Blazers stretch their winning streak to five by edging out the Phoenix Suns 96-93.
JJ Hickson also shone for Portland with 19 points and 15 rebounds.
David West led the Indiana Pacers to an unlikely comeback win as they made light of a 22-point deficit to beat the New Orleans Hornets 81-75.
The Hornets led 46-24 at one point but West's 25 points ensured the Pacers powered back and the Hornets lost for the 11th game in a row.
The Charlotte Bobcats fell to their 15th consecutive defeat, losing 110-88 to the Denver Nuggets, while the Washington Wizards lost their seventh straight for a league-worst 3-22 with Charlie Villanueva scoring 19 points in a 96-97 win for the Detroit Pistons.
James Harden led the Houston Rockets to their third straight win with 31 points in a 121-96 victory over the Memphis Grizzlies, while the Cleveland Cavaliers were 94-82 winners over the Milwaukee Bucks.