The Wizards trailed by 14 at the half but turned it on after the break to zip past the Heat, while LeBron James sat out the fourth quarter for the defending champions.
Jan Vesely had 11 rebounds for Washington.
James felt his team's sloppy play in the third quarter -- which included seven turnovers -- set the tone for Washington's comeback.
"We came out in the third quarter and turned the ball over too much," James said. "That allowed them to get back in the game. We can learn from every game, no matter if it's preseason."
The Heat welcomed back Mario Chalmers and Udonis Haslem, who each missed the opening six preseason games because of quadriceps strains. Chalmers finished with eight points and five assists and Haslem had eight rebounds.
Bosh and Wade each went 5 of 6 from the floor in the opening quarter and James was 3 for 6 with three rebounds and two assists for Miami, which opened the game on a 12-1 run.
"It was the first time for all of us to get out there together," Wade said. "To come out with that start, it felt good. "
The Los Angeles Lakers selected Croatian center Ivica Zubac with the No. 2 pick in the second round (No. 32 overall) of the 2016 NBA draft on Thursday night.
Myron Medcalf and Amin Elhassan discuss the 76ers' selection of Furkan Korkmaz with the 26th overall pick.
The Boston Celtics drafted Michigan State forward Deyonta Davis, sending him to the Memphis Grizzlies in a draft night trade that involves future draft picks.