Johnson scored the game-winner in the 82nd minute after goals in the 26th and 38th minutes, as the Wizards rallied from a 2-0 deficit.
Davy Arnaud scored the other goal in the 25th minute, and also had one assist, while goalkeeper David Hartman made four saves for the Wizards (5-2-1), who have lost only once in their last six games.
They improved to 7
New England's Taylor Twellman scored his MLS-leading seventh goal of the season.
New England was denied in its attempt to extend its unbeaten streak to eight straight games.
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