After Everton's defeat by Manchester City on Saturday made the Gunners' top-four spot secure, the France striker grabbed the only goal of the game, rising in front of marker Craig Dawson to convert Santi Cazorla's corner.
Baggies keeper Ben Foster had no chance with that effort but the England World Cup hopeful was on his toes minutes later to push away Carzola's effort as it headed for the bottom corner.
Arsenal went close once more, after the break, when German Lukas Podolski hit a half-volley from 10 yards that bounced off the post and into the arms of Foster.
The Baggies' best effort of the match came in the first half - a Graham Dorrans 25-yard shot that Wojciech Szczesny patted down.
Arsenal finish their league campaign at Norwich before an FA Cup final against Hull at Wembley on May 17th.