The Merseysiders did not have it all their own way in Sunday's 3-1 triumph at Fulham, as substitutes Kevin Mirallas and Steven Naismith eventually settled matters with a goal apiece in the final 11 minutes.
Everton are now on their longest top-flight winning streak since 2002 and sit just four points behind fourth-placed Arsenal, who they host next Sunday, with a game in hand.
For Martinez, the goal of achieving a top-four finish is very much on the cards.
"You work hard over eight or nine months to see where you're going to be in the final eight games of the season," the Everton manager said.
"I think now it's fair to say with the amount of points we have that we've got a realistic chance to fight for that aim.
"I'm not saying we're going to get there or not but in the same manner we admire Arsenal, we know that they've been getting consistently in the Champions League and that develops a real understanding of what to do in these sort of games.
"So for us the challenge against Arsenal on Sunday is phenomenal. It's a welcome one but again it reflects what a good season we've had so far."
Rock-bottom Fulham were more than a match for Everton and looked set to earn a point when Ashkan Dejagah fired home an excellent 71st-minute equaliser to cancel out David Stockdale's own goal.
But Mirallas and Naismith had other ideas, taking the game beyond Fulham late on.
And Martinez confessed that his side did not begin the contest as well as he had hoped they would.
"Fulham started the game better than us," the Spaniard added. "I thought they were very energetic, very strong in the first half, and they stopped us completely in what we wanted to do on the ball.
"At that moment we had to work very hard to defend, to be strong off the ball, and I think we did that really well.
"The second half it was a little bit different. I thought the fresh legs of Steven Naismith, Kevin Mirallas and Aiden McGeady gave us a spark and probably we were the team finishing stronger.
"But even then it was a really open game, both teams trying to win and creating many chances. I felt defensively we were full of character and a team that it is prepared to fight and give everything to try to get a positive result. So from a manager's point of view I am as happy as can be in a performance."
Midfielder Ross Barkley made way for Naismith at the break and Martinez revealed that it was an enforced change.
Martinez explained: "He has a knock on his calf. We expect him to recover quickly and hopefully he will be available for Sunday."
