He needed 18 points to reach the milestone and came away with 25, as well as 10 assists and seven rebounds. It helped the Heat improve to 25-12, the best record in the Eastern Conference.
Aged 28 years and 17 days, James beat the previous record-holder - Kobe Bryant of the Los Angeles Lakers - by more than a year.
Meanwhile, Russell Westbrook scored 32 points as the Oklahoma City Thunder never trailed, beating the Denver Nuggets 117-97. As the Thunder improved their league-best record to 31-8 with their fifth straight win, Kevins Martin and Durant added 20 points apiece.
Also in action was Austin Rivers, who got one over on his dad as he and the New Orleans Hornets beat the Boston Celtics - coached by Doc Rivers - 90-78. Rivers added eight points off the bench, whilst Al-Farouq Aminu led the Hornets with 18 points and nine rebounds.
Jeff Teague secured 28 points and 11 assists to help the Atlanta Hawks to a 109-95 win over the Brooklyn Nets. Zaza Pachulia had 13 points, 11 rebounds and eight assists, finishing two dishes short of a triple-double.
The Chicago Bulls claimed a 107-105 win over the Toronto Raptors in overtime, with Carlos Boozer pouring in 36 points and pulling down 12 rebounds.
With three seconds of the extra period remaining, Luol Deng, who had 19 points and seven assists, hit the winning jump shot.
Tim Duncan scored 19 points whilst Tony Parker had 17 points and 11 assists to lead the San Antonio Spurs to a 103-82 win over the Memphis Grizzlies.
Elsewhere, DeMarcus Cousins secured 21 points and 16 rebounds as the Sacramento Kings edged out Washington 95-94 to end the Wizards' three-game winning streak. Kyrie Irving's 31 points sent the Cleveland Cavaliers to a 93-88 victory over the Portland Trail Blazers.
The Orlando Magic beat the Indiana Pacers 97-86, whilst the Dallas Mavericks were 105-100 winners over the Houston Rockets.