Miami overcame a three-point deficit to start the fourth quarter and put the game away with Allen's three free throws over the final 16 seconds.
James missed the game with hamstring problems, Bosh was battling an illness and Wade was out for the seventh time in nine contests with an ankle injury.
Miami took 41 of its 75 shots from 3-point range and made 17 of them. Rashard Lewis added 17 points for Miami while Shane Battier and Mike Miller combined for nine of the team's 3-point field goals. Battier had 15 points and Miller contributed 14.
Washington lost its third in a row and has dropped 50 for the season. The Wizards were led by 23 points from A.J. Price.