Jake Guentzel scored the lone goal in the shootout for the Lightning, who trailed 5-1 in the second period but responded with four straight goals to tie the game, including two on a 5-on-3 power play in 23 seconds.
Brandon Hagel had a goal and two assists, Darren Raddysh had a goal and an assist, and Guentzel had two assists for the Lightning (35-14-4), who have won three straight. Andrei Vasilevskiy made 29 saves.
Tampa Bay was hosting its first NHL outdoor game, and did so in front of 64,617 fans on a chilly night (41.8 degrees Fahrenheit at puck drop).
Morgan Geekie had two goals and an assist, and Charlie McAvoy had two assists for the Bruins (32-20-4), who extended their point streak to six games (4-0-2). Alex Steeves, Matthew Poitras and Viktor Arvidsson scored. Jeremy Swayman made 41 saves.
Oliver Bjorkstrand got Tampa Bay within 5-2 with a power-play goal during a goal-mouth scramble at 10:28 of the second period.
Raddysh finished a pass from Kucherov with a one-timer from between the circles at 15:50 to make it 5-3, and Nick Paul scored just 23 seconds later at 16:13 on a tap-in from a pass from Guentzel to cut the deficit to 5-4.
Kucherov then scored on a one-timer from the right circle to tie the game 5-5 at 11:50 of the third.
Hagel scored the fastest goal to start an outdoor game in NHL history to give the Lightning a 1-0 lead 11 seconds into the game with a snap shot from the high slot. Tampa Bay won the opening face-off and the puck eventually went to Kucherov, who weaved into the offensive zone before getting the puck to Hagel.
The Bruins then scored four straight goals.
Steeves tied the game 1-1 at 11:24 after a turnover by Paul in his defensive zone came to Michael Eyssimont, who did a spin move before finding Steeves in the slot for a one-timer.
Geekie gave Boston a 2-1 lead at 15:36, tipping a McAvoy shot from the half-wall.
Arvidsson pushed it to 3-1 on the power play when he deflected McAvoy's shot from in front at 18:03.
Poitras made it 4-1 at 2:22 into the second period, scoring from the slot with a backhand that went over Vasilevskiy's glove for his first goal of the season. Poitras was recalled from Providence of the American Hockey League on Jan. 29.
Geekie scored his second of the game for a 5-1 lead at 8:18 of the second. David Pastrnak faked a shot from the right circle and slid it across to Geekie for a one-timer from the left circle.
After a stoppage in play after Swayman saved a shot at 11:01 of the second period, Hagel whacked his pads and the goalie punched him several times before skating to center ice where he and Vasilevskiy exchanged punches before receiving fighting majors.
Lightning forward Anthony Cirelli sustained an injury in the first period after a hit by Mark Kastelic with 1:44 remaining and did not return.
Tampa Bay defenseman Victor Hedman returned to the lineup after missing 22 games because of elbow surgery.
