James became only the fifth NBA player to score 10 or more points in 500 consecutive games, joining an elite list of Kareem Abdul-Jabbar, Michael Jordan, Karl Malone and Moses Malone.
Dwyane Wade added 20 points as the Heat's first road win of the season saw them top 100 points for the fifth straight game.
The Indiana Pacers remain the NBA's only unbeaten team after Paul George scored 31 points to lift them to a 99-91 win over the Detroit Pistons.
CJ Watson added 15 points and David West had 12.
The Houston Rockets continued their strong start to the season as they improved to 4-1 with a 116-101 victory over the Portland Trail Blazers.
James Harden scored 33 points while Dwight Howard added 29 and 13 rebounds.
Kemba Walker's 25 points led the Charlotte Bobcats to a 102-97 victory over the New York Knicks.
Carmelo Anthony led New York with 32 points, but was 10 of 28 from the field in the process and they could not keep pace.
A Sacramento Kings rally fell short as they went down 105-100 to the Atlanta Hawks, who had led by 17 going into the final quarter.
Al Horford had 27 points and 10 rebounds to lead the Hawks.
Monta Ellis had 30 points as seven Dallas players reached double figures in an impressive 123-104 victory over the Los Angeles Lakers.
Tony Parker had 24 points as the San Antonio Spurs kept the Denver Nuggets winless this season with a 102-94 victory in Colorado.
Brook Lopez had 27 points to lead the Brooklyn Nets to a 104-88 victory over the Utah Jazz, while Eric Bledsoe's 25 points helped the Phoenix Suns beat the New Orleans Pelicans 104-98.