Pre-Season Friendly: Accrington 1 Everton 4
Marouane Fellaini was named among Everton's starting XI, with England internationals Phil Jagielka and Leon Osman also included in a strong team.
And it did not take long for the visitors' Premier League quality to show, as Ireland international midfielder Gibson opened the scoring on 11 minutes before striker Anichebe headed a cross from Mirallas past Accrington keeper Ian Dunbavin five minutes later.
Steven Naismith then had an effort cleared off the line as Everton threatened to overrun their League Two hosts, before Belgium international Mirallas got in on the act with a fine curling free-kick that made it 3-0.
However, Accrington pulled a goal back almost immediately through Peter Murphy, who converted a Nicky Hunt cross to reduce the hosts' deficit before the break.
Martinez made wholesale changes at the interval, with Steven Pienaar and Leighton Baines among those brought into action.
But it was Osman who extended Everton's lead just after the hour when he nodded home Seamus Coleman's centre.
Martinez's side continued to dominate proceedings but did not add any more goals. Their next pre-season match is sure to be a more testing affair as they travel to Championship side Blackburn on July 27.