Midfielder Barry has made an instant impact at Goodison Park with some impressive performances in the centre of the park following his loan move from Manchester City.
Martinez offered Barry the opportunity to play first-team football after a big recruitment drive from new City manager Manuel Pellegrini ensured he slipped down the pecking order at the Etihad Stadium.
The 32-year-old is relishing a new challenge on Merseyside and has settled in his new surroundings more easily than he did at City following his move from Aston Villa back in 2009.
He told the club's official website: "It's only the second transfer in my career. Certainly at City it took me a lot longer to feel part of the team and part of the club but here it has felt that bit different,
"Everybody has welcomed me in and the manager has suited my style, so I think that played a big part in me fitting straight in. Obviously positive results help as well.
"It's been a mixture of things but I've felt part of Everton straight away."
Barry also feels appreciated by the Everton fans, who have been treated to some attractive football since Martinez replaced David Moyes.
He added: "The job I do is an unspectacular one so maybe the Everton fans have the ability to look at the game a bit differently to fans at other clubs and appreciate what I do,
"But it's great to hear that you're being appreciated for the job you're doing. I enjoy that sort of role.
"It doesn't hit the headlines as much as your attacking midfielders or your strikers, but for me it's about doing a job and if the team is successful I'm reaping the benefits of that."
