Everton opened up a 2-0 lead at Goodison Park thanks to first-half goals from Seamus Coleman and Steven Naismith, and they looked in little trouble of surrendering their advantage for much of the second half.
However, Aaron Ramsey reduced Arsenal's arrears with seven minutes remaining and Olivier Giroud headed in a last-minute equaliser.
It was a second straight 2-2 draw for Everton, following their opening-day point at Leicester.
Martinez was delighted with what he saw in the first half, but put his players' failure to hold on for victory down to fatigue.
"The performance was exactly as we wanted," he said. "The first-half performance is what we want to be doing.
"But we ran out of legs a little bit and Arsenal, as you would expect, threw everything forward and you get into a position where the third goal is going to be vital.
"At 2-0 up, we were so comfortable in the first half. Tim Howard never had to make a save and we were always a real threat.
"The performance in the first half was exactly what we wanted. The result... at this stage of the season, we'll take it.
"Obviously it's a disappointment to drop points but we were really happy with the performance and the way that we played Arsenal in the first half."