Walcott won the penalty for Robin Van Perie's opener and then slotted home the second inside the first 11 minutes of the 3-0 victory at Molineux.
Walcott said: "I think that's what we want to do. When we come away from home we need to be right on the front foot.
"The early goal sort of settled our nerves a bit and we could get the ball down, pass it. The high line came into my hands really, it helped me score my first goal as well.
"These are the games where we need to win."
Arsenal are now five points clear of fourth-placed rivals Tottenham Hotspur.