The 25-year-old will join on a free transfer after his contract at Shakhtar Donetsk ran out earlier this summer.
The Blues have paid a loan fee of just over €2m for the midfielder.
Valverde had made it clear that Mina's World Cup had not forced him to change his opinion on the player and with six centre-backs as well as six non-EU players at the club, Barca were always likely to sell.More news: Trump going ahead with taxes on $16 billion in Chinese imports
Bernard has signed a deal until 2022 at Goodison Park, and revealed the key role played by Toffees boss Marco Silva, telling evertontv: "I chose Everton because of all the things I had heard about Marco Silva and, after speaking to him, I was happy with what he said to me".
"Everton is a club with a great structure and a lot of tradition, historically the fourth most successful club in England".
Both clubs are in the market to add another centre-back to their squad before the transfer window for Premier League clubs to make signings slams shut on Thursday at 5pm. "I believe I will be able to show my best as a player here".
"The fact he has 14 caps for Brazil is a further reflection of his quality".
Everton will sign their second defender from Barcelona of the summer.