Bergeron gave the Bruins a 3-2 lead 1:39 into the third when he scored off a pass from Boyes after a Tampa Bay turnover.
Wayne Primeau also scored for the Bruins, who lost 8-3 at Florida on Friday night in their season opener. Hannu Toivonen stopped 29 shots.
Ruslan Fedotenko and Ryan Craig scored for Tampa Bay, and Marc Denis made 22 saves in the Lightning's home opener.
The Lightning battled back from a 2-0 deficit for the second consecutive game when Fedotenko and Craig scored second-period goals. Tampa Bay trailed 2-0 after the first Thursday night in a 3-2 shootout win in Atlanta.
Primeau put the Bruins up 1-0 at 5:24 of the first.
Boyes made it 2-0 when he redirected Zdeno Chara's shot at 13:19. Rookie Phil Kessel, taken fifth overall in the 2006 draft, assisted on the goal for his first career point.
Notes: Chara had two assists. ... Boston D Milan Jurcina (bruised elbow) missed his second straight game. ... Tampa Bay had won its three previous home openers, outscoring the opposition 15-4. ... Bruins RW Shean Donovan played in his 700th NHL game. He had an assist.