Everton, Middlesbrough managers united in Arteta praise
"He gives us a composure," Moyes enthused after the game. "Not only that, he can eliminate players. He can play in the middle of the park, as you saw. He can go past players in the wide areas. He can pass it as well when need be.
"At times we gain free-kicks from it because he does very well getting his body between the ball and the opposition."
Moyes acknowledged Arteta has proven the difference this season between winning and losing games.
Moyes said: "The creativity Mikel gave us today has certainly been missing."
He added: "Today I thought we made chances. Aston Villa last weekend, we probably had as many passes but we didn't create any chances.
"Today I thought we did make some chances. Albeit, Middlesbrough made chances as well.
"The games have been tight. Man Utd here, nothing in the game. The Villa game, nothing in the game.
"Today, it was a bit like that as well but we got the goals at the right times."
Those goals came courtesy of free-scoring defender Joleon Lescott and on-loan South Africa midfielder Steven Pienaar's first in English football.
Boro boss Gareth Southgate also applauded the performance of Arteta but felt poor defending was more responsible for the goals than the Spaniard's good play.
"I don't know how much of a role he had in the first goal but obviously it was a poor goal for us to concede," Southgate said.
"The second wasn't too clever either.
"He's an outstanding footballer - there's no question about that.
"But whether that was the difference between the sides, I'm not sure."