Boston pulled even at 3 when Boychuk snapped a 20-game goal drought 8:11 into the third. The Bruins have outscored their opponents 78-43 during the third period this season.
Filppula put the Lightning ahead 3-2 from the slot at 9:31 of the second after the teams had each scored twice earlier in the period.
Palat ended Tampa Bay's long scoring drought against the Bruins on a short-handed goal to make it 1-0 just 27 seconds into the second. The Lightning went 169:55 without scoring after Filppula, in the first of four games between the teams this season on Oct. 3, had a goal midway through the second in that game.
Boston had outscored Tampa Bay 11-1 in winning the previous three meetings.
After Barberio scored at 4:13 of the second, the Bruins had two goals in 1:31 by Paille (6:58) and Soderberg (8:29) to tie it at 2.
Paille beat Bishop with a backhander between his legs before Soderberg had a breakaway goal.
Lightning star Steven Stamkos played in his second game since returning from a broken right shin, which occurred in a game at Boston on Nov. 11. The centre, who has no points through the two games, lost out on a breakaway chance in overtime when time ran out as he approached the Boston zone.
NOTES: Boston RW Jarome Iginla played in his 1,295th NHL game, moving him past Pierre Turgeon into sole possession of 56th place on the career list. ... Chicago senior adviser Scotty Bowman has resumed scouting games after having right knee replacement surgery on Jan. 24. The 13-time Stanley Cup winner is a regular at Lightning home games. ... Tampa Bay RW Ryan Callahan, acquired from the New York Rangers on Wednesday for Martin St. Louis, assisted on Barberio's goal for his first point with the Lightning.