Matthew Tkachuk and Aleksander Barkov scored, and Bobrovsky made 26 saves for Florida (48-24-6), which has lost nine of its past 12 games (3-7-2).
"I like the way we played," Panthers coach Paul Maurice said. "Tonight, we got better in the third period when we had nothing in the tank. To do that, they pushed through it. So I was happy with the effort, disappointed with the result."
Tkachuk gave the Panthers a 1-0 lead 37 seconds into the game, taking advantage of an odd bounce off the end boards when the puck went to him on the inside of the right circle.
McAvoy tied it 1-1 at 5:42 of the first period, beating Bobrovsky from the high slot.
"There's a certain simplicity that we have to play [with], but it allows us to have success," McAvoy said. "At times, it might be boring, but it's winning hockey. So it's getting the puck behind them, staying above them, being opportunistic but overall, not beating ourselves."
The Bruins failed to capitalize on three power-play opportunities in the first period, but Coyle scored on the man advantage at 15:45 of the second period, redirecting Brad Marchand's centering pass at the doorstep to give Boston a 2-1 lead.
Boston was 1-for-22 on the power play in its past eight games before the goal.
"You can get a big goal on the power play, it can be huge," Coyle said. "Especially come playoff time. So it's a work in progress. There's things we can do better, but of course it's nice to see it go in."
Barkov tied it 2-2 at 5:24 of the third period, scoring off a rebound of Sam Reinhart's shot from the left side with the teams skating at 4-on-4.
"I think our effort was really good," Barkov said. "They got the lead there in the [second period], but we just stick with it. We played how we wanted to play and [Bobrovsky] was really good in the net, so I mean, of course you want to win these games, but came up short today."
NOTES: Boqvist's goal was his first overtime goal in the NHL. He has 34 goals in 232 regular-season games...Boston forward Danton Heinen had an assist and has 10 points (five goals, five assists) in his past 11 games...McAvoy set a career high with his 11th goal of the season, surpassing the 10 he scored in 2021-22.