Five wins in a row have taken Everton to within four points of Arsenal - with a game in hand.
And Martinez is hopeful his side can push the Gunners all the way in the race for a Champions League place - with Arsene Wenger's men due at Goodison Park on Sunday.
"I think it will go down to the last day," said the Spaniard.
"Nobody will stop fighting. Every game in this League is down to small margins. With the amount of points that we have, we have a realistic chance to fight for that [top four] aim.
"We admire Arsenal and know how they've got in to the Champions League for the last 18 years, or whatever. That gives an understanding of what to do in these games. Arsenal looking over their shoulder is a great compliment for us, but we are looking over our shoulder as well.
"There may be more pressure on them, but they cope with that expectation. When you start the season, there are six football clubs with the budget to cope with that expectation. They need to be coping with that expectation.
"We always embrace the challenge. We really enjoy playing at Goodison. We've created an incredible relationship between fans and players. The fans have been patient, we feel supported and feel we've got a real advantage playing at home.
"We've got Arsenal, Manchester United, Manchester City and Crystal Palace at home, and they are as difficult as they can get. But even then we feel if we can be perfect in what we do, we can get maximum points.
"The intensity of every game is important and it doesn't matter if you are fighting to avoid relegation, win a title or finish in the top four - it's the same. You need to gain momentum and find a real focus in the squad. I wouldn't say [it would be a] 'miracle', but football miracles happen."
Martinez is delighted with how his side are hitting form and is particularly pleased with how the whole squad are contributing.
"The biggest compliment I can pay my players is that I can select any XI from our squad and be confident that we can play the way I want to play and show the concepts we have as a team," he added.
"I always believe that we are going to be as good as the squad. The days that you can rely on 11 players are gone. That is how you judge at the end of the season how successful you can be as a team. The fresh legs of Steven Naismith and at the right time with Kevin Mirallas and Aiden McGeady gave us a different spark."