The Senators won the opener of the Eastern Conference quarterfinal series, which will continue on Friday in Montreal.
Montreal owned a 2-1 lead heading into the third, but both Silfverberg and Methot put slap shots past Carey Price. Silfverberg's goal came just 3:27 into the period.
The game was marred by a series injury suffered by Montreal's Lars Eller in the second period. He was struck by a blind-side hit to the head from Ottawa's Eric Gryba, who received a game misconduct penalty.
Eller bled heavily while lying on the ice and eventually had to be placed on a stretcher.
The Canadiens and Senators are meeting in the playoffs for the first time. Montreal posted a 14-7-3 home record during the regular season while Ottawa had a losing record on the road.
Price, who lost six out of seven starts late in the season, gave up the four Ottawa goals on 31 shots.
Anderson turned aside a barrage of 48 shots as he tries to help his team advance in the playoffs for the first time in his career. He suffered all four losses in first-round setbacks with Colorado in 2010 and with Ottawa last year.