Laynce Nix and Ian Desmond each collected singles with no outs in the ninth off Edward Mujica (9-6), and two batters later Morse pounded a 1-1 pitch deep into the left field stands, erasing a 4-3 deficit.
Rick Ankiel drove in a pair of runs and Desmond finished 2-for-4 with and RBI double for the Nationals.
Atahualpa Severino (1-0) got the win with a scoreless eighth in relief and Henry Rodriguez picked up his second save for Washington.
Logan Morrison went 2-for-4 with a solo homer for the Marlins, who lost in the first contest of their final three-game series of year -- and their last games at Sun Life Stadium.
The team, to be renamed the Miami Marlins, will open a new stadium in downtown Miami next April.