The 23-year-old, who was brought to Wigan Athletic on loan by Roberto Martinez in January, has signed a five-year contract with Everton.
|14/1||Everton are 14/1 with Paddy Power to finish in the top four
Joel, who started for Wigan in the club's FA Cup final win over Manchester City, kept four clean sheets in his 12 outings for the Latics and conceded 17 goals.
"He arrives to challenge Everton's established custodian Tim Howard for a place in Roberto Martinez's starting line-up and is travelling with the squad to their Austria training camp," read a statement from Everton.
Earlier on Tuesday, Everton also completed a move for defender Antolin Alcaraz, who signed a two-year deal after his contract with Wigan expired at the end of June.