The brace took the Dutchman to 100 goals for the London club - a credible achievement given he has started just 175 games since moving from Feyenoord seven years ago.
Van Persie, a player who has had a bad-boy image in the past, took over from Cesc Fabregas as captain this summer and Wenger has been pleased with the Dutchman's leadership.
"Robin is a man who speaks his mind," Wenger said.
"I think he is one of these guys who develops with responsibilities.
"You would think he is not a natural captain at the start but he really has grown well into the role. I have observed him and he does well there."