Koehler (4-10) allowed two runs on three hits over six innings to earn the win, which dropped the Nationals from playoff contention in the National League East and handed the crown to the Atlanta Braves.
Giancarlo Stanton belted a two-run homer, Christian Yelich added a solo shot and Donovan Solano added two hits and an RBI for the Marlins.
Washington's Dan Haren (9-14) absorbed the loss, surrendering three runs on eight hits over six innings.
Ryan Zimmerman clouted a two-run homer for the Nationals, who trail Cincinnati by 5 1/2 games for the final NL wild card spot with seven games to play.
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'