Twenty-eight points apiece from James and Bosh saw the Heat move third on the all-time list for consecutive wins in a single season as they overtook the 1970-71 Bucks.
There was a scare for the Heat when Dwayne Wade appeared to suffer a neck injury early on, but he later returned to the game.
The Los Angeles Lakers overcame the continuing injury woes of Kobe Bryant to beat the Indiana Pacers 99-93.
Bryant played despite an ankle injury, but managed only one quarter and had no impact on the game.
Dwight Howard led the Lakers in his absence with 20 points and 12 rebounds.
Kevin Durant scored 26 points to help the Oklahoma City Thunder to a 117-104 win over the Orlando Magic.
Russell Westbrook added 23 for the Thunder, who scored 73 points in the first half.
James Harden poured in 37 points and pulled down seven rebounds as the Houston Rockets overcame the Minnesota Timberwolves 108-100.
Jeremy Lin added 24 points for the Rockets, who trailed by 20 points in the third quarter.
Luol Deng led Chicago with 23 points as the Bulls saw off the Golden State Warriors 113-95.
John Wall scored 29 points in the Washington Wizards' 96-87 win over the New Orleans Hornets and Rudy Gay's 28 helped the Toronto Raptors to a 92-78 win over the Charlotte Bobcats.
The Atlanta Hawks beat the Phoenix Suns 107-94, the Denver Nuggets saw off the Memphis Grizzlies 87-80 and the Dallas Mavericks were 96-86 winners over the Cleveland Cavaliers.