Bergeron became the fourth Boston player and 94th in NHL history to reach the milestone with the second assist on Brad Marchand’s goal at 15:08 that gave the Bruins a 4-1 lead.
Marchand immediately pointed at Bergeron after scoring his goal and the Boston bench emptied to celebrate with the 37-year-old center along the boards.
The other Boston players to reach 1,000 points are Ray Bourque (1,506), Johnny Bucyk (1,339) and Phil Esposito (1,012), who was at the game as a Lightning radio broadcaster. Bergeron is the 41st player to have 1,000 points with one franchise.
Boston needs two wins to match the 1929-30 Bruins squad that started the season a franchise-best 19-2.
Tampa Bay had won four in a row.
Foligno and Coyle scored 31 seconds apart in the second to put Boston up 3-1.
Foligno, who took a shot off the chin during the morning skate, had a rebound power-play goal at 5:07.
Coyle was credited with his in-close goal after a video review. It was originally called no-goal when the referees ruled the net had come loose before the puck cleared the goal-line.
Pastrnak, on the power play, and Balcers traded goals 11 seconds apart early in the third before Paul's man-advantage goal made it 5-3 at 10:38.
Pastrnak has 13 goals in 15 games against Vasilevskiy, and extended his point streak to seven games (four goals, 10 points).
Paul and Krejci had first-period goals. Krejci has four goals during a three-game goal streak.
Tampa Bay outshot 9-0 over the first eight minutes of the game.
Boston’s David Pastrnak and Jake DeBrusk collided in the offensive zone and wound up on the ice during late first-period power play. DeBrusk skated to the bench afterwards but was OK and finished with two asists.
Bruins: C Trent Frederic left with an upper-body injury.
Lightning: C Ross Colton (upper body) was scratched.
Bruins: Play Wednesday night at Florida.
Lightning: Host St. Louis on Friday night.
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Mark Didtler, The Associated Press