Douglas-Roberts took an inbound feed from Josh McRoberts at the top of the key, drove inside the paint and floated the shot in as the buzzer sounded.
With the win, the Bobcats remain in the hunt for the Eastern Conference's No. 6 seed. Charlotte trails Washington by one game for the sixth spot as each team wraps up its respective season on Wednesday.
The Bobcats host the Chicago Bulls and the Wizards travel to Boston for a matchup with the Celtics.
Charlotte holds the tiebreaker over the Wizards because of their 3-1 record in the season series.
Charlotte's Kemba Walker, who returned from a two-game absence with a groin injury, scored just five points with seven assists on 1-for-9 shooting.
Jeff Teague, the reigning Eastern Conference Player of the Week, tallied 11 points with five assists for the Hawks.