Bergeron scored a power-play goal on a wrist shot from near the right face-off dot for the Bruins (30-17-8), who are 3-0-3 in their past six games.
The Bruins honored Bergeron, who had a goal and two assists, in a pregame ceremony for playing in 1,000 NHL games. He hit the milestone Tuesday against the New York Islanders.
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"To get the win was very special," Bergeron said. "That's what I said on the bench, just we've got to find a way, and we did. Obviously we'll take, I guess, that bounce getting the power play in overtime. But we stuck with it and got it done."
Charlie McAvoy, Danton Heinen and Brad Marchand each had a goal and an assist, David Krejci scored, and Tuukka Rask made 25 saves to extend his personal point streak to 10 games (8-0-2) for Boston.
Alex Iafollo, Anze Kopitar, Nate Thompson and Oscar Fantenberg scored, and Jonathan Quick made 28 saves for the Kings (23-27-5), who had won their previous three games.
"In the end it was a good point," Kings coach Willie Desjardins said. "The guys battled back ... We could've been better at the start of the third. But the thing I've liked about this group is that they don't give up and it was good to see them turn around and get a couple and end up with one point anyway."
McAvoy scored on a tip from the top of the crease at 4:38 of the second period to give the Bruins a 1-0 lead. Bergeron set up the goal with a pass across the slot.