The former Trinidad and Tobago international will assist U21 manager Alan Stubbs and ensure information is filtered down to coaches throughout the academy set-up.
"I've had three fantastic seasons with Wigan and I wouldn't take anything away," said Lawrence. "It's a very good club, I got on really well with the people there.
"But then obviously when the call comes from the manager himself, I've got a good relationship with him.
"I worked with him as a player, I've worked with him as a coach, and I've got a bit of loyalty to him because he was the one who got me involved in coaching in the first place."
Lawrence said that his belief in Martinez's attacking philosophy convinced him to move to Goodison Park.
He said: "In football there's no wrong or right way in terms of the way you want to play the game but I think the way the manager sees the game being played is the same as I would like to see.
"Hopefully I can bring my experience from playing because some players do learn quicker than others and it's important you recognise the ones who might be picking up on things a bit quicker and help them understand what the manager wants."
Martinez has already signed Arouna Kone, Antolin Alcaraz and Joel Robles from Wigan, as well as four Latics backroom staff.
Andy Holden, who was part of former manager David Moyes' backroom staff, has left the club.