The Toffees ended a run of 21 years without a win at Old Trafford in midweek and are now looking to break a similar, if slightly shorter, sequence on the Gunners' home turf.
Graham Stuart and Andrei Kanchelskis were the goalscorers when Everton last won at Arsenal in January 1996 but Martinez has brought a belief to the club - strengthened by the win over United - which raises hopes of an end to their misery in north London.
"I don't think we've played with fear at all this season," said Martinez, who has won at the Emirates with Wigan in April 2012.
"I felt when we went to City we didn't have the right belief in our performance but I think we showed that at Old Trafford.
"Arsenal deserve a lot of respect because if you win 11 of your first 14 games you are the outstanding team in the competition.
"We are looking at ourselves to try to use this as a terrific test to see how quickly we can improve and challenge these sides.
"The Emirates Stadium is one of the hardest places to go and be yourself.
"Arsenal as an attacking team are as good as it gets and their movement and forward play requires maximum concentration in everything you do.
"From our point of view we are ready for that test. We are excited about challenging the best team in the league so far."
In Everton's favour ahead of Sunday's game is their defensive record, which is the top-flight's equal best with Southampton at 13 in 14 matches.
They have kept five clean sheets in their last six matches with only Liverpool having breached them in the 3-3 Merseyside derby.
However, from open play Everton have not conceded since Yannick Sagbo scored for Hull on October 19 and Martinez said that shows how well the team is working.
"The old cliche that defenders are there to defend and attackers are there to attack doesn't work any more," said the Spaniard.
"In the modern game as a team you defend with 11 players and you attack with 11 players and it has been important that we work as a unit.
"Everyone is really clear in the roles we have and at the moment we have terrific focus and intensity in our performances and that shows not only at the back but going forward.
"We went to Old Trafford and we had a left-back in the 18-yard box scoring in the 86th minute - that shows the responsibility the players take in both roles."
Martinez said that team ethic extended beyond the starting XI, citing the case of Nikica Jelavic.
The Croatia international has started just two league matches since Romelu Lukaku arrived on loan: one of those was against Chelsea when the Belgium international was ineligible to face his parent club and the other was the following week at West Ham when he was replaced at half-time by Lukaku who went on to score the winner.
"He has been working incredibly hard behind the scenes," said the manager.
"Every player wants to start, you need to have that in your nature otherwise you will never be a professional footballer.
"But in the same manner I think it has been key in pushing other players to be at their best.
"He has been setting really good standards in training and he is a vital player for us.
"People from outside look in and think it is about the players who start but there is a bigger picture than that.
"Bryan Oviedo (who has scored twice in as many appearances standing in for the injured Leighton Baines) has been a really good example for any footballer who needs to be ready behind the scenes.
"When you get the opportunity there are no excuses. Niki is in the same boat and is ready to help the team when needed."
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