Barzal is on a career-best, five-game goal streak, and he has six goals and two assists during that span.
Vancouver goaltender Collin Delia made 30 saves, while Elias Pettersson, Oliver Ekman-Larsson and Ilya Mikheyev each collected two assists.
Ty Smith also scored for Pittsburgh, which lost its sixth straight game, just one off matching its longest losing streak of the season.
Nicolas Aube-Kubel, Martin Fehervary and Garnet Hathaway also scored, Dylan Strome had three assists, and Darcy Kuemper made 25 saves.
Washington has won a franchise-record seven straight road games and is 7-0-2 in its past nine games overall and 12-2-3 in its past 17.