The striker scored his seventh goal for the club and first since mid-October when he beat his marker to power home Santi Cazorla’s cross on 60 minutes.
Arsene Wenger’s side, who sealed their first victory in three Premier League games, had a string of chances to take a first-half lead, with Welbeck going closest.
West Brom only came to life in the last 15 minutes and Saido Berahino had their best chance to equalise when he headed against the crossbar late on.
Welbeck returned from injury after missing the midweek Champions League victory over Borussia Dortmund, as Olivier Giroud made his first league start of the season and Laurent Koscielny also returned.
And the former Manchester United striker looked lively from a wide-right position, latching on to Cazorla’s neat pass midway through the first half only to see his shot well saved by Ben Foster.
Soon after the keeper was robbed by Giroud, only for the striker’s cross shot to fly across the face of an empty goal.
The visitors were guilty of trying to be too precise but were being granted too much space by the Albion midfield as they tried to pick their way through.
And Foster was needed a number of times in the opening period to deny Cazorla and Alexis Sanchez, but his team failed to test opposite number Damian Martinez before the break and struggled to threaten at all in the final third.
Arsenal suffered another injury when Nacho Monreal was forced off and replaced by Kieran Gibbs but they stayed in command in the second half, taking the lead after a brilliant run and cross from Cazorla.
It was all the Gunners deserved and they remained in their comfort zone with little to concern them until the final 10 minutes when Albion mounted a muted rally.
West Brom have won just once at home all season and Berahino’s late header against the goalframe was the only time they truly threatened.
The final whistle came as a relief to the Arsenal manager on a day when a banner in the visiting fans’ stand was calling for him to leave – ‘Arsene – thanks for the memories, but it’s time to say goodbye’ – with Gunners fans still frustrated with their team.
West Brom: Foster (6), Wisdom (6), Dawson (6), Lescott (7), Pocognoli (6), Mulumbu (5), Gardner (7), Dorrans (6), Sessegnon (5), Brunt (6), Berahino (7)
Used subs: Anichebe (5), Gamboa (5), Samaras (6)
Arsenal: Martinez (7), Chambers (7), Mertesacker (7), Koscielny (7), Monreal (6), Ramsey (7), Flamini (7), Sanchez (8), Cazorla (9), Welbeck (8), Giroud (7)
Used subs: Gibbs (7), Oxlade-Chamberlain (6)
Man of the Match: Santi Cazorla