McCoughtry scored 18 points in the second half to rally her team.
The Mystics (5-7) led much of the first three quarters. Washington, the least effective 3-point shooting team in the league, went 7-for-10 from beyond the arc en route to a 39-34 halftime lead.
But the Mystics went cold for nearly 6 minutes in the fourth quarter, and the Dream (8-3) took advantage with a 12-0 run.
Tiffany Hayes added 11 points for Atlanta, and Erika de Souza scored 10 and grabbed eight rebounds.
Mystics rookie point guard Bria Hartley led her team with 16 points and forward Emma Meesseman added 15.
Washington went 3 for 4 at the free-throw line, while the home team was 20 for 23.
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