Dyche (left) was one of the main contenders to replace Lampard (right) along with Marceloa BielsaEverton have named Sean Dyche as their new manager on a two-and-a-half-year contract until June 2025.
The Toffees sacked Frank Lampard last week with the club 19th in the Premier League and two points from safety.
Dyche, 51, had been out of work since he was dismissed by Burnley last April after almost 10 years at the Clarets.
He is Everton majority owner Farhad Moshiri's seventh managerial appointment since buying into the club in February 2016.
Dyche, whose first game in charge will be at home against league leaders Arsenal in the top flight on 4 February, said in a statement it was "an honour to become Everton manager".
He added: "My staff and I are ready and eager to help get this great club back on track.
"I know about Everton's passionate fanbase and how precious this club is to them. We're ready to work and ready to give them what they want.
"That starts with sweat on the shirt, effort and getting back to some of the basic principles of what Everton Football Club has stood for for a long time.
"We want to bring back a good feeling. We need the fans, we need unity and we need everyone aligned. That starts with us as staff and players.
"Our aim is to put out a team that works, that fights and wears the badge with pride. The connection with the fans can then grow very quickly because they're so passionate.
"There is quality in this squad. But we have to make them shine. That's the job of me and my staff."
Dyche will be supported at Goodison Park by former Nottingham Forest winger Ian Woan as his assistant manager and ex-England international Steve Stone as first team coach.
Chairman Bill Kenwright said Dyche quickly convinced the board he had "exactly the right attributes to make himself a great Everton manager".
Dyche guided Burnley to two promotions from the Championship and two top-half Premier League finishes, including a first European qualification in 51 years, after they finished seventh in 2017-18.
Everton have 18 games remaining and after hosting Arsenal, their next game is away at Merseyside rivals Liverpool on 13 February.
Former Leeds United manager Marcelo Bielsa was also a contender for the job and held talks with Everton's hierarchy.
