The Toffees controlled the first half, before Mesut Ozil found the breakthrough for the Gunners with 10 minutes remaining.
However, Barcelona loanee Gerard Deulofeu - a second-half substitute - was on hand to restore parity four minutes later and Martinez believes that his team deserved the equaliser.
Speaking to BBC Sport, the Spaniard said: "The first half performance was as good as it can be from an away side at the Emirates, to take the sting out of the game and stop Arsenal playing and creating a threat.
"Their goal came from nothing but to display the character and come back to score with a goal of that quality was fitting for a game of that standard."
The draw leaves the North London outfit five points clear of Liverpool at the summit of the Premier League, while Everton remain fifth.