James had 17 points in the final period for the Heat, who held the short-handed Knicks to eight field goals in the second half and will go for the sweep Sunday.
Dwyane Wade added 20 points for the Heat and Mario Chalmers had 19, hitting consecutive 3-pointers in the fourth quarter.
Carmelo Anthony scored 22 points on 7-of-23 shooting for the Knicks, who are playing without Amare Stoudemire, Jeremy Lin and Iman Shumpert.
The Knicks broke the record set by Memphis from 2004-06. They haven't won a playoff game since April 29, 2001, Game 3 of a best-of-five series against Toronto.
THUNDER 95, MAVERICKS 79
DALLAS (AP) — Kevin Durant had 31 points on 11-of-15 shooting and Oklahoma City never trailed in taking a 3-0 series lead over the defending NBA champs.
Game 4 is Saturday in Dallas.
Russell Westbrook added 20 points for the Thunder, while Serge Ibaka had 10 points and 11 rebounds and James Harden and Derek Fisher both had 10 points.
Dirk Nowitzki had 17 points to lead Dallas, and Jason Kidd had 12.
The Mavericks have a nearly impossible climb to avoid being the first defending champion knocked out in the first round of the playoffs since Miami five years ago. After beating Dallas in the 2006 NBA finals, the Heat were swept in four games by Chicago in the opening round of the Eastern Conference playoffs the next season.
No NBA team has ever come back from a 3-0 deficit to win a best-of-seven series.