The 53-year-old Wilson appeared at a Tuesday news conference at Toronto's Air Canada Centre after signing a four-year deal.
Wilson was fired May 12 by San Jose after the Sharks lost to the Dallas Stars in the second round of the playoffs. He succeeds Paul Maurice, who was fired by Toronto last month after missing the playoffs in back-to-back seasons.
"I'm the happiest person in the National Hockey League today," Wilson said. "I'm fulfilling a dream.
"Obviously if you lose your job in the league you often wonder if you'll get another opportunity," he said. "The furthest thing from my mind at any point in my hockey career was that I would have an opportunity -- ever -- to coach a team like the Toronto Maple Leafs with their incredible history."
"I think he's everything we need here in Toronto," interim GM Cliff Fletcher said. "He's one of the top coaches in the league."
He was also the only candidate Fletcher interviewed for the job.
The Maple Leafs finished 12th in the Eastern Conference with a 36-35-11 record last season, missing a playoff spot by 11 points.
Wilson is the 20th different coach the Leafs have had, including interim ones, since they last won the Stanley Cup in 1967.
"I view the Leafs as a team in transition," he said. "It's going to require a lot of work and a lot of attention to detail.
"As a coach, you just want to put yourself in a position to knock on the door," Wilson said. "And if we can be knocking on the door in two years, I think that will be a tremendous accomplishment. Then we've just got to knock the door down."