Martinez, 39, led Wigan to a memorable FA Cup final triumph last month and despite failing to save them from relegation to the Championship, he emerged as the outright first choice of Everton chairman Bill Kenwright from a three-man shortlist believed to also include Porto's Vitor Pereira and former Schalke coach Ralf Rangnick.
Kenwright made an official approach on May 24 and four days later Martinez told Wigan chairman Dave Whelan he wanted to leave.
The two clubs agreed a compensation package believed to be £1.5m earlier this week.
Martinez compensation agreed
A statement on the Wigan website read: "Roberto's influence at the club, both as a player and manager, an association that spanned over a decade, will remain unforgettable to all Latics fans and the club would like to place on record its appreciation for his tireless work during that time, culminating in FA Cup glory last month at Wembley Stadium.
"The club now begins its search for a new manager as we seek to make a return to the Barclays Premier League and progress in the Europa League."
Former Everton striker Kevin Campbell believes Martinez will be "a breath of fresh air" after David Moyes' more regimented approach.
"The way he plays the game, the fans are going to like the style, and I think the players are going to improve, and that's really important," he said.
"Everton have wonderfully gifted players and I think he will give them a new lease of life."
"We will always be friends for the rest of our lives, he is just such a wonderful character but he is going to a great club with great supporters. I wish him the very best of luck."
Dave Whelan
Wigan chairman Dave Whelan is sorry to see Martinez move on, but is delighted the Spaniard has gone to Everton after confirming there were no issues in agreeing a deal with his Toffees counterpart Kenwright.
Whelan told Sky Sports News: "It's something I accepted when he asked me if he could leave.
"I did expect Everton to come and get his services. They've done a wonderful move, Roberto is, and has been for us, one fantastic manager.
"We will always be friends for the rest of our lives, he is just such a wonderful character but he is going to a great club with great supporters. I wish him the very best of luck.
"There was never any disagreement between Bill and myself. We agreed everything from the start, it was how it was put across to various agencies and how people picked it up.
"Bill and myself always had the agreement that they would pay the compensation, and Roberto could be interviewed and could be their manager.
"I'm sorry to lose Roberto, he's a great manager but I'm delighted he has gone to Everton."
