Harden made six three-pointers and scored 40 or more points for the 24th time this season, The Associated Press reported.
Houston entered the day in fifth place in the Western Conference standings, a game behind of fourth-place Oklahoma City.
Eric Gordon added 32 points and had eight three-pointers.
Boston has lost five of their last six. Kyrie Irving led the Celtics with 24 points, nine rebounds and six assists. Al Horford finished with 19 points.
The Celtics did a decent job trying to contain Harden, limiting him to 14-of-31 shooting from the field and nine free throw attempts.
But Boston was inefficient when it had the ball, shooting 48 percent (37-of-77) for the game and 29 percent (8-of-28) from beyond the arc.
The Celtics started the fourth quarter with a 17-6 run to cut what had been a 28-point deficit to 106-96 with 6:08 to play.
Terry Rozier got fouled on a three-pointer at the five-minute mark, but missed all three free throws.
It was 109-100 when Nene gave Houston a second opportunity after a Gordon miss. It wound up in Harden’s hands and he stepped back and swished his sixth triple of the game to push the lead back up to 12.
But leading 112-104 Harden fouled out when he was called for an offensive foul with 1:44 left.
The Celtics came up empty on their next trip down the floor and they were forced to foul after failing to corral Austin Rivers’ miss from deep.