The 18-year-old has impressed since breaking into the Arsenal first team and scored his first senior goal for the club in the 2-1 win at Swansea last month.
That came a week after he made his full Barclays Premier League debut in the win over Stoke.
Czech Republic forward Rosicky has been thrilled with Gnabry's contribution and anticipates further progress under boss Arsene Wenger.
"He has a manager here who will develop him and who has a history of developing young players," Rosicky told arsenal.com.
"Nothing better could happen to him than being offered a new contract here.
"He is a good dribbler and can take on you on in the one-on-one situations. That can be his strong point in the future and for an 18-year-old he is doing very well.
"He is a very talented boy, 18 years old and he still has plenty of things to learn and he will find no better place to learn than here. I think it works very well for both sides that he stays here.
"He's got a great environment for his growth and the potential and talent is there, so let's see how he will develop."
Gnabry, who signed his first professional contract last year after shining in the club's youth ranks, made his Arsenal debut in the 6-1 win over Coventry in last season's Capital One Cup.
His Premier League bow came as a substitute in the defeat to Norwich in October 2012 and four days later he came off the bench in the Champions League clash with Schalke.
In doing so, the former Stuttgart youngster became the second youngest Arsenal player to appear in both competitions, after Jack Wilshere.
Wenger has high hopes for the speedy teenager.
"We signed Serge Gnabry because I believe that he will be a great player," Wenger said on arsenal.com.
"He's nearly there now, and competes for a regular place in the team in the Premier League.
"He has technical ability, he has power, he has pace, he has finishing quality and he can give a final ball. He's a powerful offensive player and I believe that he will be a great player for Arsenal football club."
Gnabry said: "Arsenal is a great football club and I'm so happy to be here and sign a new contract. For me, now it's all about working hard and developing my game.
"I've had some good opportunities recently in the first team, which has given me extra confidence, and this will only improve my game. I'm looking forward to giving my best for this club in the years to come."
Gnabry is likely to be involved in Tuesday night's Capital One Cup fourth-round tie with Chelsea, if he can shake off an ankle injury.
Isaac Hayden and Ryo Miyaichi are also set to feature, but midfielder Mathieu Flamini (groin) and Mikel Arteta (suspended) are out.
Wenger is uncertain how long Arsenal must do without Flamini.
The Gunners boss said: "He has a scan today (Monday) and then we will get the results as to how long he will be out.
"It doesn't look like a bad muscle injury but honestly it's too early to give you a precise answer. It's a groin injury."
Arteta was dismissed, unjustly in the Spain midfielder's opinion, for denying Marouane Chamakh a goalscoring opportunity in Saturday's win at Crystal Palace.
"He is suspended for one game, so I will not appeal," Wenger said.