Van Persie was described as the "most complete striker" in the Premier League by manager Arsene Wenger last week.
Walcott told arsenal.com: "He's been absolutely sensational since he took over the armband from Cesc Fabregas. He's been a great captain for us already. The free kicks, the goals, his general play he's enjoying his football. It's nice to see him fit, that's very important to have him fit all season.
"I was gutted when that turn and chip didn't go in, I was praying for that to go in and I was already celebrating because it would have been the best goal I've ever seen. We want Robin to be firing on all cylinders and he is starting to show the true player he is."