Van Persie inspires Gunners as they hit three against Baggies.
Van Persie netted the first and then turned provider for both Thomas Vermaelen and Mikel Arteta.
Aaron Ramsey's exquisite through-ball helped the Arsenal skipper net his 11th league goal of the campaign.
Vermaelen then netted his first goal for almost two years as he turned in a Van Persie cross and, after the break, Arteta fired home after neat work between the Dutchman and substitute Tomas Rosicky.
The Baggies were undone in the 22nd minute by an exquisite pass from Ramsey.
The Wales international's low ball cut through the West Brom defence and straight to Walcott, whose blocked shot fell kindly for Van Persie to net his 29th goal.
Wenger's side were still struggling to regularly test Ben Foster, but managed to double their advantage six minutes before the break. Alex Song's ball to the back post was volleyed across goal by Van Persie, allowing Vermaelen to fire home his first of the season.
With less than 20 minutes remaining, Wenger brought on Yossi Benayoun and Rosicky.
The move immediately had the desired effect as Arsenal scored with the next attack as Van Persie and Rosicky played a neat one-two, before the former teed up Arteta to fire home from 20 yards.