Columbus traded the forward for a first-round pick in the 2021 NHL Draft and a fourth-round pick in the 2022 NHL Draft from Toronto.
The Sharks received a fourth-round pick in 2021 from the Maple Leafs for forward Stefan Noesen.
Foligno was captain of the Blue Jackets since May 20, 2015. The 33-year-old can become an unrestricted free agent after the season. He has scored 16 points (seven goals, nine assists) in 42 games this season and 482 points (203 goals, 279 assists) in 950 regular-season games with Columbus and the Ottawa Senators.
"It was an emotional couple of days," Foligno said. "I have the utmost respect for (Columbus general manager) Jarmo [Kekalainen] and how he brought this to my attention, talking about how if there was something that made sense for the team that he would try to keep me in the loop, and that meant a lot to me. Ultimately it's always what's best for the team, but he was able to fill me in on some things and see where I was headed, and it just fell into place where it worked out to come to Toronto.
"It's been a pretty amazing day once you've made the decision, but a hard day obviously with all the emotion attached to Columbus. Now that it has sunk in and how excited I am to join this team and help in any way I can, it really gets me going."
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The Maple Leafs (28-10-3) are in first place in the seven-team Scotia North Division, six points ahead of the second-place Winnipeg Jets. The Blue Jackets (15-20-8) are seventh in the eight-team Discover Central Division, nine points behind the fourth-place Nashville Predators. The top four teams in each of the four divisions will qualify for the Stanley Cup Playoffs.
Foligno will have to quarantine for at least seven days due to COVID-19 protocols after arriving in Toronto because he is crossing the United States-Canada border.
"A lot of the players, both (general manager) Kyle [Dubas] and (coach) Sheldon [Keefe] have reached out too," Foligno said. "I think they're excited about the opportunity to get to work. You can tell how focused that group is on trying to win, so I'm looking forward to joining that. It lines up mentally with where I'm at and what I feel is necessary to win hockey games. I'm looking forward to joining a team that feels that way and has that same mindset that I've had for so long here in Columbus. I know [Jason] Spezza pretty well, know John Tavares a little bit, played with Auston [Matthews] with the [United States] team, and Joe Thornton is a legend and somebody I've respected from afar."
Tavares said Foligno's style should make him a good fit with the Maple Leafs.
"I think he's just an extremely competitive player," the center said Monday. "And I think within that he has a real good understanding of his skill set and how he makes an impact. To me, that's his ability to protect the puck, hang on to it, makes the little plays on the wall and his finishing ability in front of the net. He just seems to have that determination and drive to make an impact."
Foligno's father, Mike Foligno, played 129 games for Toronto in four seasons from 1990-94 and was part of the run to the 1993 Campbell Conference Final.
"I remember the passion of the fans and how excited my dad was," Foligno said. "After beating [the] St. Louis [Blues] (in the Norris Division Final), we got to walk the streets home, and that parade atmosphere, that kind of stuck with me. I know the reach the team has and I'll never forget that run. It's something my dad talks about to this day and it's engrained into that city.
"I'm looking forward to going on another one and being the next Foligno to do so. I'm excited about joining and helping this team."
The Maple Leafs also acquired goalie David Rittich from the Calgary Flames on Sunday for a third-round pick in the 2022 draft.
Defenseman Morgan Rielly said the additions of Foligno and Rittich should strengthen Toronto for a strong stretch run.
"It speaks volumes, but as players, we have to take that responsibility," Rielly said. "With the results we've had this year, I think that there's an opportunity for us to play some good hockey here and I think the players know that. I mean, it's time that we got working and get the process down here before playoffs."
On Monday, Toronto acquired defenseman Ben Hutton in a trade from the Anaheim Ducks for a fifth-round pick in the 2022 draft.
The Blue Jackets made three trades in three days ahead of the 2021 NHL Trade Deadline, which was 3 p.m. ET on Monday.
They sent forward Riley Nash to the Maple Leafs on Friday for a conditional seventh-round pick in 2022, and defenseman David Savard was traded to the Tampa Bay Lightning on Saturday in a three-team deal that also involved the Detroit Red Wings. Columbus received a first-round pick in 2021 and a third-round pick in 2022 in the Savard trade.
Coach John Tortorella told the Blue Jackets website on Saturday that they need to pivot after believing they were a playoff team at the start of the season.
"The core of this team has been together, and we were growing, we were getting there, and things have fallen off," Tortorella said before a 4-3 loss at home to the Chicago Blackhawks. "The business end of it comes in, players wanting to leave and stuff like that, and you know what? You need to start over. I'll be honest with you, it needs to start over because there are just a lot of different personalities that come in.
"I don't know if they fit or not. We've still got to go through the process of trying to figure out who they are, who we are. Are we going to do it together? And do you understand? Are you going to buy in? And all that stuff. I don't have the answer as far as where it goes right now. We're still trying to find out about the new personnel coming in here."
Noesen was held without a point in five games this season with the Sharks (18-18-4), who are in sixth place in the eight-team Honda West Division, four points behind the fourth-place Blues. The 28-year-old scored six points (two goals, four assists) in 12 games for the Sharks' American Hockey League affiliate, the San Jose Barracuda. He has scored 54 points (31 goals, 23 assists) in 204 regular-season games for the Sharks, Pittsburgh Penguins, New Jersey Devils and Ducks.
NHL.com staff writer Mike Zeisberger and independent correspondent Dave McCarthy contributed to this report