Bergeron, who is Bruins captain, had the secondary assist on a goal by Brad Marchand at 15:08 of the second period at Amalie Arena in Tampa.
The 37-year-old forward has 18 points (nine goals, nine assists) in 19 games this season. He has 409 goals and 591 assists in 1,235 NHL regular-season games.
Bergeron is the 94th NHL player to get 1,000 points, and 41st to do it with one team. He is the fourth Bruins player to reach the milestone after Ray Bourque (1,506), Johnny Bucyk (1,339) and Phil Esposito (1,012).