Toffoli, who also had an assist, Troy Stecher (goal, two assists) and J.T. Miller (three assists) each had three points for the Canucks (33-22-6), who are 3-1-1 in their past five games after going 0-3-1 in the previous four.
Toffoli, who was acquired in a trade from the Los Angeles Kings on Monday, made it 7-3 at 11:10 of the third period and scored his 20th goal at 13:03 to make it 8-3.
"It feels good," he said. "Obviously scoring a goal this early makes it a little easier, but at the same, you've just got to keep working. It sounds cliche, but keep trying to build these guys, not just my linemates, but everybody. Lots of talk and lots of work, and we're going to have some fun."
WATCH: [All Bruins vs. Canucks highlights]
Jacob Markstrom made 34 saves for Vancouver, which moved into second place in the Pacific Division, two points behind the Vegas Golden Knights and one ahead of the Edmonton Oilers.
"They're the best team in the League," Miller said. "The crowd was awesome. Big rivalry, they have some really good players. It's an exciting game to get up for, and I think it's a good test, obviously. We played a really good game. Everybody was going, worked hard, wore them down. It feels really good in here."