The Toffees cruised to a 3-0 victory at Brunton Park, managing to hold their nerve after reports of racial abuse from the spectators after Arouna Kone had given the visitors an early lead.
Martinez hailed his team for remaining calm throughout and creating an attacking threat against the League Two Cumbrians.
"I thought we played well and controlled the threat Carlisle could bring and don't underestimate the clean sheet against a team that can put the ball in the box very quickly," the Spaniard told BT Sport.
"Scoring three goals and creating good chances, overall it was a well-worked performance."
Regarding the matter of spectator abuse, Martinez felt that referee Lee Mason and the Cumbria police handled the situation very well by making an announcement on the ground and asking anybody to report such incidences.
"I thought it was dealt with in a very good manner," Martinez added.
"We agreed to make an announcement. I thought the referee was very good in how he dealt with that.
"It was just a reaction, something that we don't want to see. It was just a reaction to the celebration of the goal and that was it. It would be wrong to highlight such a moment in an enjoyable footballing festivity."
The Toffees will next meet Bournemouth at Dean Court on the weekend of February 20.