Arsene Wenger completed a £42.2million swoop to land the Germany international from Real Madrid on transfer deadline day.
The acquisition of 24-year-old Ozil immediately started a debate about who may find their place under threat in the Arsenal midfield, with Wenger also having Santi Cazorla, Jack Wilshere, Mikel Arteta, Tomas Rosicky, Mathieu Flamini and Ramsey among his options.
Ramsey, in particular, has made an excellent start to the season and is now showcasing his best form since suffering his horrific leg break at Stoke in February 2010.
The 22-year-old Wales international, who is preparing for World Cup qualifiers against Macedonia and Serbia, has scored three goals in his last five games and picked up two man-of-the-match awards, and is set to feature for Coleman's men despite nursing a groin problem.
The Dragons boss believes the former Cardiff starlet's performances are more than good enough to fend off any competition for a place at his club.
He said: "All Aaron has to do is keep doing what he has been doing.
"Ozil is a great player, but if Aaron keeps playing like he is, then I don't think it will be Aaron Ramsey losing his place.
"I'm not the Arsenal manager and it's not up to me. It's up to Arsene Wenger and he's a quality manager.
"But if Aaron keeps doing what he has been doing then I can't see him losing his place."
He added: "Aaron has been brilliant. He is showing now what we all thought he was capable of with his performances. He's been superb for Arsenal this season.
"All that has happened is he has taken time and games to get stronger and get that fluidity back in his play.
"He's got a spring in his step and is confident. He's doing things at both ends of the pitch. He's defending well, positionally he is much better and he's linking the play up in the last third.
"This is his best form and he's playing his best football. Even before he had the injury.
"This is the best we've seen from him."
Coleman admitted he felt it had taken Ramsey time to get over his long-term injury absence, while the former Wales captain was also heavily affected by the death of then-Dragons boss Gary Speed.
But Coleman has praised the mental strength shown by the midfielder.
"I've had serious injuries myself as a player," said Coleman, whose own playing career was ended after he suffered serious leg injuries in a car crash in 2001.
"People talk about the physical part, which is important as you have to get the agility, but mentally it is tough. It's hard.
"The best way to get through it is to keep playing lots of games and get back in the first team.
"It's been stop-start at Arsenal because he is surrounded by good players and he hasn't always been first-choice.
"It's taken him time, but you have to look at him during pre-season and the start of the season and take your hat off to him and say he's got himself back in that position where he's playing the football we all thought he was capable of.
"He's looks fantastic."