James just missed a double-double by grabbing nine rebounds in the victory, which improved the Heat to seven wins in their last nine games.
Bosh had eight boards and Miami out-rebounded Phoenix 41-36 over the course of the game. Dwyane Wade sat out the game due to a migraine headache.
Phoenix led 46-43 at halftime and was within striking distance in the fourth quarter. They trailed 87-85 midway through the period; however James scored seven points in a 9-4 run that put Miami ahead 96-89 with 1:13 to play.
Green's 3-pointer then trimmed the gap to four points, but Bosh answered with a trey of his own to make it 99-92 with 40.9 seconds remaining.