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.
"That was probably the most special thing about it,'' Bergeron said. "All the guys kind of jump on and share that with them. It's great. That's something I will remember for a long time.''
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.
"The players love him," Boston coach Jim Montgomery said. "I joked around the other day, I'm like, if he had any kind of cheat in his game he might have 1,200 points already, or 1,300. He's just so dedicated to playing the game the right way and helping the team have success."
Montgomery compared Bergeron, a five-time Selke Trophy winner as the league's top defensive forward, to Montreal great Bob Gainey.
"I think historically Bob Gainey's been the best defensive player in the game,'' Montgomery said. "I had the fortune of watching him in his hey-day with Montreal. What Patrice has done as a center, there is so much more responsibility, he might go ahead of Bob as the greatest of all-time.''
Boston needs two wins to match the 1929-30 Bruins squad that started the season a franchise-best 19-2.
"That's why they're an elite team in the league, when they waned they kept it together and stuck with it,'' Tampa Bay coach Jon Cooper. "When it didn't go our way, we didn't handle it that well. Two good teams. It was a good game for the most part, but they deserved more than we did.''
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 (4 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.
The Associated Press contributed to this story.