James Johnson had 17 points off the bench for Miami, which scored its most points in a game this season and got its first back-to-back wins.
Dion Waiters scored 16 points to help the Heat total six players in double figures in their first road victory since their season opener at Orlando.
Bradley Beal scored a season-high 34 points for Washington in his second game back from a hamstring injury.
John Wall also hit a season high with 34 points and Marcin Gortat scored 10 along with 16 rebounds. No other Wizards hit double figures on a night Washington lost starting forward Markieff Morris to a sprained right ankle in the second quarter.
After Dragic’s jumper gave Miami a 60-59 lead to end the first half, the Heat opened the third quarter on a 10-4 run, stretching their lead to 70-63 on Whiteside’s bank shot.
Later in the third, Miami built its lead to 13 after a 12-4 stretch. Dragic drove inside for the last four points, his latter layup making it 89-76 with 2:41 left in the quarter.
Washington trailed by as many as 17 in the fourth, before closing the game on a 10-1 run. Wall hit a 3-pointer to close it to 113-108 with 41 seconds to play.
But Washington couldn’t score on its next possession, and when Wall finally hit another 3 after Waiters’ free throw on the other end, there were only 0.9 seconds left.
Heat: Improved to 3-0 when scoring 100 or more points . Waiters started after testing a minor groin injury in pregame warm-ups . F Justise Winslow (wrist) and G Wayne Ellington (thigh) remained out, while F Willie Reed left early in the second quarter with a hyperextended right knee.
Wizards: Sank to 1-8 when opponents score 100 or more points, and 3-5 in games when both Wall and Beal start . C Ian Mahinmi (knee surgery) will participate in his first practice of the season Sunday after joining his team’s morning shootaround on Saturday, coach Scott Brooks said.
BOARDS IN BUNCHES
Whiteside extended his season-opening, double-digit rebound streak to 12 games despite waiting until 3:34 left in the second quarter to pick up his first. By halftime, he had six.
A SECOND HOME
With the victory, Miami is now 24-14 all-time at the Verizon Center. Its .632 winning percentage here is its highest at an opponent’s building where it has played at least 15 games.
Heat: Open a four-game season series against the 76ers in Philadelphia on Monday.
Wizards: Close a three-game homestand against Phoenix on Monday.
This story has been corrected to show that Wall had 34 points, not 31.