Barkov scored on the power play at 7:49 of the third period to give Florida a 2-1 lead. Jonathan Huberdeau found Barkov at the right face-off circle for a one-timer.
McAvoy scored on the power play at 13:35 to tie it 2-2, finishing a cross-ice pass from Brad Marchand.
"That's not really how we script it [on the power play]," McAvoy said. "We get the puck, we're trying to make plays, and that's where the skill takes over. I don't really know what I'm doing there, backdoor, but that's where I ended up. … That's where we just trust each other and let the skill take over. It was really nice to see that one go in."
Ullmark made the save against Huberdeau, Barkov and Owen Tippett in the shootout.
The Bruins went 1-for-3 on the power play, and the Panthers were 1-for-1.
NOTES: Hall is the second player from the 2010 NHL Draft to reach 600 points after Dallas Stars forward Tyler Seguin (640 in 752 games). Hall has 230 goals and 371 assists in 687 NHL games. … Marchand became the 10th player to reach 400 assists with the Bruins. He required the fifth-fewest games (811) to do so, behind Bobby Orr (437), Ray Bourque (544), Wayne Cashman (747) and David Krejci (774). ... It was Andrew Brunette's second game as Florida coach. ... Huberdeau played his 600th NHL game, joining Stephen Weiss (654) and Radek Dvorak (613) as the only players to reach the milestone with the Panthers. … With the assist on Barkov's goal, Huberdeau extended his point streak to six games (three goals, six assists), one shy of his NHL career high, which he has achieved three times, most recently from April 13-24.