Minten, playing his first NHL game in his hometown, opened the scoring for the Bruins (22-18-2) in the first period, scoring a power-play goal in front of dozens of family and friends.
Elias Lindholm also scored for Boston, David Pastrnak had two assists and Jeremy Swayman stopped 31 of the 33 shots he faced.
It was the Bruins' second straight win after six straight losses (0-4-2).
The Canucks (16-20-5) got a goal and an assist from Filip Hronek, Elias Pettersson had a goal and Kevin Lankinen made 18 saves.
Vancouver lost 4-3 to the Seattle Kraken in a shootout on Friday, but outshot the visitors 33-20 and outshot them 35-18.
Canucks: Less than 24 hours after a 2-for-3 power play against Seattle, Vancouver's special teams struggled. The home team went 1 for 6 with the man advantage and conceded goals due to 2 of 3 penalties they converted during the game.
Bruins: After Saturday morning's skate, Minten said he expects there will be 50 to 100 loved ones in the stands during the game. Dozens of fans wearing Bruins jerseys erupted in cheers as the 21-year-old center scored a halftime goal from the halfway point at 16:24.
Hronek passed to Linus Karlsson at the top of the crease and the puck hit Bruins defenseman Charlie McAvoy on the shin, sending it past Swayman for a power-play goal that made it 2-2 at 18:44 of the second period.
Vancouver will go six full weeks without a win on home ice. The Canucks last won at Rogers Arena on Dec. 6, beating the Minnesota Wild 4-2, and they won't play at the rink until Jan. 17 when they host the Edmonton Oilers.
Bruins: Visit the Kraken in Seattle on Tuesday.
Canucks: Begin a six-game road trip against the Sabers in Buffalo on Tuesday.
