The 25-year-old Ivorian winger didn't start either of Arsenal's opening two Premier League games, coming off the bench in both, but impressed in the Gunners' midweek Carabao Cup win over Leicester.
Arteta said: "I think he's come a long way. We know what the club paid for him when he joined and that was obviously an extra pressure but he's a player who's had to adapt, it's a different style.
"He has a really good attitude. We can see he has developed in a lot of phases of his game.
"He's very settled here, he's liked by the boys, he is a really good character to have around and I'm very positive that we're going to have him for many years and that he can perform at a really high level consistently."
Arteta also sees the summer arrival of Pepe's former team-mate at Lille, Gabriel, as a positive.
"I think it's always nice for them to have some compatriots next to them, it obviously makes things a lot easier in the first few months when they cannot speak the language,' he said.
"I think for both of them, they know each other and they've spent some time together now. I think it's a positive thing."