Wilson at 2:27 and Mantha at 4:02 gave Washington the lead, and Dowd scored into an empty net.
The Capitals played the final two periods without forward Alex Ovechkin, who sustained a lower-body injury in the first period.
Cam Atkinson scored twice to give Philadelphia a 3-2 lead in the second period. Claude Giroux scored, and Martin Jones made 28 saves for the Flyers, who open the regular season Friday against the Vancouver Canucks.
Ilya Samsonov made 22 saves for the Capitals, who begin the regular season against the New York Rangers on Wednesday.