Younès Belhanda put the home side ahead in the ninth minute from the spot after Vermaelen tripped the Moroccan forward on the edge of the penalty area.
But Arsène Wenger’s men soon fought back, netting two goals in as many minutes as Lukas Podolski notched up his third strike in three games and Gervinho fired home from close range to give the north Londoners the lead.
The Gunners held on in the second half amid sustained pressure from the Ligue 1 outfit, but Arteta insisted Wenger’s men did not panic despite some ropey moments in the final 25 minutes of their 2-1 win.
“We knew it was going to be tough, they were champions in France last year and that is no coincidence,” Arteta told Arsenal’s website.
“They work really hard, they have some talented players and scored the first one at home, which means it is always difficult to get back into the game.
“But we scored two brilliant goals and managed to get the result – it was vital for us to get the first three points away from home.
“You can’t panic, we knew in the last 25 minutes they were going to throw some bodies up front and create some problems. They missed a clear chance at the end but I am so glad to get the points, it is a brilliant way to start.”
The Gunners face defending English champions Manchester City at the Etihad on Sunday, and Arteta admitted it will be a difficult test for Wenger’s new-look side.
“It is going to be tough, City have not lost at home for a long time,” he added.
“It is big test, [we must] show our character. Last year we went there and were unlucky to lose. It will be tough, but they are the champions.”