Thompson had two goals and an assist, Alex Tuch had a goal and an assist, and Ukko-Pekka Luukkonen made 17 saves for the Sabres, who are the No. 1 seed from the Atlantic Division.
David Pastrnak and Morgan Geekie each had a goal and two assists for the Bruins, who are the first wild card from the East. Jeremy Swayman made 34 saves.
Game 2 of the best-of-7 series will be here on Tuesday (7:30 p.m. ET; ESPN, NESN, MSG-B, TVAS2, SN360).
“It's a seven-game series,” Geekie said. “Everyone is going to say you can't really lose the series in Game 1, but we put ourselves behind the eight ball, and we knew they were going to come out hard. They're a good team over there. So, I think it is important kind of get back on the horse and just continue to build our game. It's a long series.”
Thompson started the Sabres' comeback by cutting it to 2-1 at 12:02 of the third period. He got control of a loose puck behind the net and wrapped it inside the right post past the left skate of Swayman, who was looking the other way.
Thompson then tied it 2-2 at 15:44, scoring five-hole from below the left circle after Tuch forced a turnover behind the net by Andrew Peeke.
“I think those two goals are exactly what we wanted to do all night, put pucks behind them, pound their 'D,' make it a tough night on them and just hit the repeat button on that,” Thompson said. “You go back for enough pucks, it's pretty exhausting. Create loose pucks and get chances from there, and fortunately they went in for us.”