Arsenal led 2-0 at the break through Van Persie’s tap-in and a Thomas Vermaelen shot from Van Persie’s pass, despite not being at their fluent best.
But the Gunners dominated the second period against a West Brom side who only managed one effort on target.
And Mikel Arteta’s fine strike sealed the victory after a sweeping move.
Arteta’s classy 15-yard effort into the bottom corner, after some fine build-up play which twice involved the hugely impressive Van Persie, was the least the hosts deserved for their second-half dominance.
While the Baggies played with plenty of effort and endeavour throughout, and could consider themselves unfortunate to be trailing 2-0 to the only two chances of the first period, they can have no argument with the final result.
Lone forward Simon Cox was lively and dangerous but lacked support and their only effort on target did not come until the 86th minute when Steven Reid’s header was easily saved by home keeper Wojciech Szczesny.
Graham Dorrans also twice scuffed low efforts wide after the interval but the second half was a procession of Arsenal attacks as West Brom’s solid resistance was broken.
The platform was laid before the break when Van Persie followed up from close range after Theo Walcott’s initial shot from Arteta’s pass was saved.
And, following Alex Song’s cross-field pass, the excellent Vermaelen added a second from Van Persie’s exquisite cut-back.
The Gunners, whose last league outing was against Chelsea, completely dominated after the interval with Aaron Ramsey and Arteta bossing the midfield.
Belgian centre-half Vermaelen – buoyed by his first goal in almost two years – nearly made it 3-0 soon after the break but his header from Van Persie’s corner, went a fraction wide.