Minor (9-10) turned in his third straight start of allowing one run or less, giving up five hits and two walks. He struck out three.
Clinging to a 2-1 lead, Atlanta broke it open with three runs in the seventh, capped by Dan Uggla's two-run single.
Gio Gonzalez (19-8) came up short in his bid to become the major leagues' first 20-game winner, yielding a pair of runs on four hits over five frames.
Ian Desmond went 2-for-4 with an RBI for the Nationals.
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