Tiffany Hayes added 12 points for the Dream (2-2), who trailed 49-38 at the end of the third quarter before scoring a season-high 26 points in the fourth.
Shoni Schimmel hit two 3s to spark a 17-6 run, which was capped by a 3-pointer by Roneeka Hodges and pulled Atlanta within one with 4:41 to play.
Sancho Lyttle's steal set up McCoughtry's 3 which gave the Dream a 62-61 lead, their first since midway through the second quarter. Two free throws by McCoughtry capped the scoring.
Ivory Latta rimmed out a potential tying 3-pointer in the final second.
Emma Meesseman had 21 points and grabbed nine rebounds for the Mystics (2-1) while Tayler Hill added 15 points.