Pinch-hitter Bryce Harper drew a one-out walk from from Bosox closer Alfredo Aceves (0-4) and came home one out later when Bernadina lined a 2-2 pitch into the right-field corner for the go-ahead RBI.
Danny Espinoza finished 2-for-4 with a two-run double and Ian Desmond added a pair of hits for the surging Nats, who notched their sixth win in eight games and completed a three-game interleague sweep at Fenway Park.
Tom Gorzelanny (2-1) was credited with the win for tossing a scoreless eighth inning in relief of starter Jordan Zimmerman while Tyler Clippard racked up his eighth save for the National League East leaders.
Boston starter Jon Lester struck out nine in seven innings, allowing three runs on six hits while walking two.
David Ortiz went 3-for-4 with a solo homer as the Red Sox lost for the sixth time in seven contests.