The German winger, in only his second Premier League start, found the net early in the second half with a composed finish after a typically intricate build-up involving Bacary Sagna, Olivier Giroud, Jack Wilshire and Aaron Ramsey.
Wenger voiced his satisfaction in Gnabry's performance in the wake of a win which saw Arsenal climb two points clear at the top of the Premier League after six games.
"I feel he has justified my faith," the Frenchman told Arsenal's official website.
"This boy has potential, he has personality on the pitch and purpose, and he has shown that again so overall I feel it's very positive."
Wenger earlier told reporters: "He has the talent, but he has the personality on the pitch as well.
"He doesn't look timid, he looks to play with belief and, of course, talent. Unfortunately he is another player for Germany, he's not English.
"He was one-year-old when I came to Arsenal, I can't believe it. Now I have some players in my squad who were not born when I arrived at Arsenal."
Gnabry follows in the illustrious footsteps of former Gunner Cesc Fabregas, who scored his first league goal for the club at the tender age of 17 years and 113 days.