Sancho Lyttle scored 10 of her 15 points in the fourth quarter and Tiffany Hayes had 14 for the Dream (7-3), who turned Washington's 21 turnovers into 21 points.
Washington trailed by 11 in the first half and tied the game three times after halftime, but never took the lead. Sanchez' scoring fueled a decisive 11-3 run starting with 3:48 remaining.
Atlanta outrebounded Washington 17-4 on the offense boards and held an 18-0 advantage in second-chance points.
Bria Hartley scored 13 points for the Mystics (5-6), who have lost four of five.