Carlos Gomez and Jean Segura also drove in runs during the inning, which broke a 2-2 tie game.
Lucroy finished with two RBI as the Brewers won for the seventh time in eight contests and stayed within 2 1/2 games of the St. Louis Cardinals for the National League's second wild card spot.
Brandon Kintzler (3-0) was credited with the victory after hurling a scoreless sixth.
Ryan Mattheus (5-3) gave up three runs in the seventh to take the loss for the National League East-leading Nationals.
Jayson Werth and Chad Tracy knocked in runs for Washington.