Trailing 9-7 entering their final at-bat, the Nationals got Mark DeRosa aboard with a one-out walk off John Axford. Morse followed by lining an 0-2 pitch the opposite way, just clearing the right-field fence for a two-run shot.
Then in the 11th, Bryce Harper led off with a walk off Jose Veras (3-4) and went to third on Ryan Zimmerman's single. That brought up Morse, who scorched a shot into left field to score the pair for an 11-9 lead.
Tyler Clippard made the advantage hold up in the bottom of the 11th despite surrendering a solo homer to Corey Hart, collecting his 20th save as the National League East-leading Nationals completed a 6-1 road trip.
Washington scored four times in the eighth to erase a 7-3 deficit, highlighted by a two-run clout from Roger Bernadina.
Steve Lombardozzi went 3-for-5 with a solo homer and two RBI and Craig Stammen (4-1) earned the victory in the slugfest.