Martinez made five changes from the side who drew 1-1 with Stoke City in their last Premier League encounter but the Merseysiders had no difficulty in brushing aside their Championship opponents, led by two goals from Nikica Jelavic, setting up a fourth-round trip to Stevenage.
"We had big senior players out of the side," the Spaniard told the club's official website. "When that role is fulfilled by youngsters it is important they do it with responsibility and enjoy their football, and I thought that was the case.
"Joel Robles came in with Bryan Oviedo and there was the partnership with John Stones and Antolin Alcaraz [at centre-back].
|9/1||Everton are 9/1 with William Hill to win the FA Cup|
"It's the competition for places that we need. That strength in the squad will allow us to have a confident outlook and we have got a lot of people who we can rely on."
The win kept alive Everton's hopes of winning the FA Cup for the first time since 1995 but Martinez, who lifted the trophy with Wigan last season, is keeping his feet on the ground.
"We always work towards perfection but there is a long way to go and we want to keep developing," he added. "The pleasing aspect is that there are sensational signs - the way we are playing in the final third, the way we are controlling games and the youngsters fulfilling big roles.
"But football is very simple. We move on from here and then concentrate on the Norwich game [in the Premier League on Saturday]. We will take every game as it comes."