An encouraging start to the season has seen the Toffees take 19 points from nine matches to sit sixth in the table ahead of the weekend's games and still within striking distance of the top.
No-one is suggesting the Spaniard's side are capable of mounting a title challenge but Martinez believes if they can stay in touch with the top four into the new year it gives them a chance of bridging the gap.
On Sunday Everton kick off that spell at home to Tottenham, one of just four clubs currently above them who they will face in the 14 matches until the end of January - although they do play Liverpool twice and go to Manchester United, who are four points behind them.
"In any season you want to be in a strong position in the final third of the season," said Martinez.
"We have been very pleased with the start of the season and the way the squad is shaping up, now it is a matter of getting through the middle third of the campaign and seeing where that takes us.
"It is in our DNA to win trophies and leagues and we need to use that to our advantage.
"You always work towards that. How long is it going to take? It could be a season, but remember that there is a huge difference between us and the group of teams working at a certain financial level.
"But I always believe in football it is not always about finances. You just need the time to bridge that gap you have.
"We are not financially fighting with the clubs who are spending to win the league or get into the Champions League but I believe with the characters we have at the club we can challenge that but it will take time to bridge the gap."
As the Toffees have not been - and will not be - able to compete on the same financial footing as the top-flight's big six, Martinez knows he must find other ways.
He believes it can be achieved with a solid club structure, sound footballing philosophy and the development of youth.
"It can be done but you need time, you need to be patient and brave as a club and give opportunities to youngsters," he added.
"That takes composure to be able to see a player growing and allow him to give you that extra edge.
"When you spend the money you don't have to see the player developing, you can see his full potential.
"We are working well with the youngsters and the recruitment on that side.
"I am proud of the players we have here and we should aim high and be as good as we can be in that middle third."
Martinez does not believe his side have fully reached their optimum level yet but thinks that is more about putting together consistent results than the way they are playing.
"I don't think we have fulfilled our potential yet," said the Spaniard.
"We are getting stronger with a clearer understanding of what we want to do and it is about how far we can go and how quickly we can fulfil our potential, which I believe is huge.
"We have new players in the team and we have certain aspects of our play we want to do in a different manner that is going to take a little bit of time.
"We have that competitive edge, we have a team who can compete against anyone, but when you are trying to implement certain football concepts you need to practise and master them and that takes time and that is where we are.
"One thing is the way we want to play, the style, the philosophy and we are there; the other thing is being extremely good at it and that is where you need to keep working hard and anticipating things before you see them.
"That is where being together for a long time allows you to be stronger."