The Spaniard left Goodison Park for the Gunners in a £10million deal on the final day of the summer transfer window last year after six years playing for the Scot.
Moyes admits when he first brought in Arteta from Real Sociedad, after a two-year spell with Rangers, he was unsure how well he would fare.
However, he now sees Arteta as a key player and one of the leaders in Arsene Wenger's side, the visitors on Wednesday.
"I'm not surprised. He is a very gifted footballer, he is a great lad to work with," said Moyes.
"I think the manager (Wenger) was probably surprised what he offers: he is a great professional, a really good footballer and I'll be delighted to see him back - just disappointed it is in an Arsenal jersey.
"I see Mikel now and when they are having a bit of a huddle he is the one who is the leader and I think they look to him, he has taken on a lot of the leadership qualities in that side.
"He was part of the rebuilding process that had to take place at Everton when we tried to develop the team and move it on.
"We took Mikel on loan from Real Sociedad after he'd been at Rangers and it hadn't quite worked for him.
"We were a bit unsure when he first came here. He was a bit lightweight to play in the middle and so we played him off the side and he was great there for us, he really was.
"Eventually he moved on to be a real key figure in that period and was a big part of us developing and moving Everton forward."
Arsenal were the last team to beat the Toffees at Goodison in the league, last March.
Since then they are 10 unbeaten on home turf and although a run of one win in their last seven matches has curtailed what was a promising start to the campaign, Moyes is confident it has given them a platform on which to build their season.
"We are going to try to push," added the Scot, whose fifth-placed team are just a point ahead of the Gunners.
"A club like Everton are not out of their depth in Europe and we have to try to push to do that as we've missed out the last couple of years.
"Arsenal are a good team and that is why they continually finish in a Champions League spot so it is a tough game.
"We have been playing quite well and no-one gets an easy time at Goodison."
Earlier in the season Moyes had tried to play down his side's start to the season by insisting they be judged after 10 matches.
Now after 13 league fixtures - which included five draws in the last seven - he insists a longer-term view needs to be applied to the team's achievements.
"I think you measure Everton over the period of 10 years of what they have done," he said.
"If you measured us last year you would say we were rubbish this time last year.
"And this time last season Arsenal were not doing very well and they went on to finish in the top four."