Sunderland won just their second Championship match of the season by beating Burton Albion on Saturday.
Sunderland have been woeful in the Championship so far this season, and currently sit in the relegation zone.
However, Barton has backed the Black Cats to get out of trouble - as he feels Coleman is a very astute appointment at the Stadium of Light.
Barton said: “Sunderland will improve under Cookie now. I don’t think they’ll get promoted, but they won’t go down.”
Coleman has already made an impact at Sunderland, after he guided them to just their second league victory of the season at the weekend.
Sunderland beat Burton Albion 2-0 at the Pirelli Stadium in their latest outing, to relieve some of the pressure building around the Stadium of Light.
Sunderland have now moved off the bottom of the Championship table, and there is some optimism building at the Wearside club.
Sunderland are next in action at the weekend, when they take on Reading.