James also grabbed nine rebounds, made five steals and passed off three assists Tuesday while Bosh contributed eight rebounds for Miami.
The Heat bounced back from a loss three nights earlier at Utah and defeated the Suns for an eighth meeting in a row.
Miami improved to 36-14, the second-best record in the Eastern Conference to league-leader Indiana (40-11).
The Suns, who won six of their prior eight games, slipped to 30-21, seventh in the Western Conference.
Miami will visit Golden State on Wednesday before a break for the NBA All-Star game, at which James, Bosh and star guard Dwyane Wade — who did not play because of a migraine — will represent the Eastern Conference.
Mario Chalmers added 13 points and Norris Cole came off the bench to score 10 points for Miami.
Gerald Green led the Suns with 26 points while Goran Dragic and Channing Frye added 15 points each and P.J. Tucker netted 14 for Phoenix.
Miami led 92-89 when James sparked a closing 11-8 run with a jump shot and a slam dunk. After Green and Bosh traded 3-pointers, James added four free throws in the final seconds to clinch the triumph.