A 3-2 defeat against Crystal Palace at Goodison Park on Wednesday means Everton passed on the opportunity to leapfrog back over Arsenal – who beat West Ham 3-1 24 hours prior - into fourth in the battle to secure qualification to next season's UEFA Champions League.
With four league games remaining, Arsene Wenger's men will secure a top-four finish if they match Everton's results over the coming weeks.
And Martinez has cranked up the pressure by suggesting that the London club will not win all their remaining fixtures, insisting his Everton side, a point further back, will fight to the end.
"We have 12 points to play for," the Spaniard said.
"We are close to fourth place and we are still in a good position.
"I don't think Arsenal will take maximum points from their remaining games.
"If you looked at the start of the season, it's a magnificent position.
"The way we finished the second half gives me great optimism we can finish the season strongly."
Goals from Jason Puncheon, Scott Dann and Cameron Jerome prevented Everton from making it eight home league wins in a row.
Steven Naismith and Kevin Mirallas netted for the hosts to make it 2-1 and 3-1 respectively.
Martinez knows it is a fixture that Everton were tipped to win, but paid tribute to a plucky Palace performance.
"The result is a hurtful one because of the expectation that we had and the desire that we brought into the game," he said.
"In the first half we played with a bit of wanting to win so much that we forgot about simple basics. In the second half, it was completely different.
"We were positive and back to being ourselves.
"But in every aspect, the game played into Crystal Palace's hands. They were well organised, happy to sit behind and play on the counter and dead balls."
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