The Sabres received forward Anders Bjork and a second-round pick in the 2021 NHL Draft.
The Bruins also got forward Curtis Lazar.
Hall said he could be in the lineup when Boston hosts Buffalo on Tuesday (7 p.m. ET; NESN, MSG-B, NHL.TV).
"I'm available tomorrow night for sure," Hall said. "I'm going to be driving from Buffalo to Boston today and should be arriving around 7 (p.m. ET), something like that. It's been a team -- since I knew that I would be traded for the last few weeks -- it was a team that I really wanted to join and really wanted to be a part of."
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Hall signed a one-year, $8 million contract with the Sabres on Oct. 11 and can become an unrestricted free agent after this season.
The 29-year-old forward scored 19 points (two goals, 17 assists) in 37 games for the Sabres. He hasn't played in a game since April 3, having been a healthy scratch the past four as a precaution.
"For myself, obviously, it wasn't the season that I would have like to have. Not even close," Hall said. "These last few days, you do some soul-searching and you look back on what you could do better and look forward to the future. And obviously I'm so happy and excited to be a part of the Bruins. I can't wait to play and I can't wait to be part of a group like that."
The No. 1 pick by the Edmonton Oilers in the 2010 NHL Draft, Hall has scored 582 points (220 goals, 362 assists) in 664 regular-season games for the Sabres, New Jersey Devils, Arizona Coyotes and Oilers.
Hall won the Hart Trophy voted as the NHL most valuable player in 2017-18, when he scored an NHL career-high 93 points (39 goals, 54 assists) in 76 games for the Devils.
He was traded from New Jersey to Arizona on Dec. 16, 2019, and scored 27 points (10 goals, 17 assists) in 35 games with the Coyotes before signing with the Sabres.
"I don't want to set expectations too high," Hall said. "I just want to come and win games. And I feel like I can help teams do that. Unfortunately so far in my career it hasn't happened. But a fresh start, especially after the season that I've had and everything like that, I think the last five or six days being away from hockey and being able to kind of reflect, I still believe in myself a lot as a player. I feel like I can add things to a team. I still feel like I can really do that. That's what makes me most excited."
Hall has scored 12 points (four goals, eight assists) in 14 Stanley Cup Playoff games.
Boston (21-12-6) is fourth in the eight-team MassMutual East Division, four points ahead of the New York Rangers and Philadelphia Flyers; Buffalo (10-25-6) is last. The top four teams will qualify for the Stanley Cup Playoffs.
"Ultimately, things didn't go as well as I would have liked and obviously our team was in the situation that it's in," Hall said. "A bit of a circuitous route to get here, but looking at it in the summer, this is a team in Boston that I would have loved to play for and I'm here now, so I'm very excited about that."
Lazar has not played since March 31 because of a lower-body injury. The 26-year-old scored nine points (five goals, four assists) in 33 games for Buffalo this season, and has 70 points (25 goals, 45 assists) in 317 games with the Sabres, Calgary Flames and Ottawa Senators.
Lazar has one season left on the two-year, $1.6 million contract ($800,000 average annual value) he signed with Buffalo on Sept. 24.
"I always loved playing Boston just because it's going to be high-paced, it's going to be gritty," Lazar said. "They kind of have that swag and that physicality, which is one area of the game that I really pride myself in, is taking the body and setting the tone that way.I kind of see myself as a perfect fit on that team just in that regard."
Bjork scored five points (two goals, three assists) in 30 games for the Bruins this season. The 24-year-old has 39 points (16 goals, 23 assists) in 138 games for Boston, which selected him in the fifth round (No. 146) of the 2014 NHL Draft.
Bjork has two seasons left on the three-year, $4.8 million contract ($1.6 million AAV) he signed with the Bruins on July 29.
The Bruins acquired defenseman Mike Reilly in a trade from the Senators on Sunday for a second-round pick in the 2022 NHL Draft. The 27-year-old is in the final season of a two-year contract he signed with the Montreal Canadiens on June 26, 2019.
"Obviously they've got the big line (Brad Marchand, Patrice Bergeron, David Pastrnak) and some new faces in there for sure from years past, but I think I can help break the puck out and first and foremost, defend, play hard and then work itself out from there," Reilly said. "Try to be the fourth guy on the rush and make plays, be support with these skillful forwards. So definitely excited for that."
NHL.com staff writer Amalie Benjamin and independent correspondent Matt Kalman contributed to this report