The 22-year-old has made four appearances since joining on loan from Real Madrid, where he struggled for starts, in the January transfer window.
He caught the eye in Thursday's 1-1 draw against Benfica in the Europa League, with no Arsenal player creating more big chances (two) than the midfielder, level with Hector Bellerin.
Odegaard also led the way for passes in the opposition half (29) and tackles (three), highlighting two different aspects to the Norway international's game.
Arteta has been impressed by what he has seen so far and has backed the loanee to grow in stature over the coming weeks.
"On the pitch he is showing a lot of character and intelligence - ingredients any leader should have," Arteta said. "He's been here a very short time but has adapted really quickly.