Keane, 24, moves to Goodison Park after agreeing a five-year contract.
The England defender is Everton's fifth signing of the summer, with Malaga striker Sandro Ramirez also joining on Monday for £5.2m.
Those signings take Everton's spending in this transfer window to more than £90m, a total including a deal worth up to £30m for goalkeeper Jordan Pickford.
"Michael was high among the players we wanted to bring to Everton this summer and I'm delighted he is here," said Everton boss Ronald Koeman.
"We knew there would be competition from other clubs but Michael believes in the ambition of Everton and what we want to do and this is the perfect moment for him to come."
Keane added: "I watched a lot of Everton last season and I believe I really suit how the team likes to play.
"I feel like this is a great place to come and continue my development as a player."
Keane started his career at Manchester United but was sold to Burnley by then manager Louis van Gaal for £2m in 2015. United will profit from his move to Everton as they inserted a 25% sell-on clause in the transfer.
Everton have signed England Under-21 goalkeeper Pickford from Sunderland and brought in Ajax captain Davy Klaassen in a move worth almost £24m.
Nigeria striker Henry Onyekuru arrived from KAS Eupen in Belgium for £7m - although he will spend the next 12 months on loan at Anderlecht.
Spanish striker Sandro, 21, has signed a four-year deal at the club after they met the player's £5.2m buyout clause.
He scored 14 goals for Malaga in La Liga in 2016-17, having joined them from Barcelona last summer.
This is not expected to be the end of Everton's big spending, with Koeman determined to test Swansea City's resolve to keep 27-year-old Iceland midfielder Gylfi Sigurdsson.
They could also conduct further business with Burnley this summer, with reports talks have opened to sign striker Andre Gray.
Everton have signed Michael Keane from Burnley for a fee that could rise to a club record-equalling £30m.