The England striker nodded home on the hour to down the Baggies and secure a first victory in three Premier League outings for Arsene Wenger's side, lifting them into the top four
Consecutive defeats domestically - following a 3-3 draw with Anderlecht in the UEFA Champions League, which saw Arsenal throw away a three-goal lead - had seen Arsene Wenger's ability to guide the London club to further silverware this season questioned.
However, a 2-0 win over Borussia Dortmund in midweek proved Arsenal's capabilities against European opposition and the grit displayed at The Hawthorns will have enthused Wenger.
Arsenal were always in control of the contest and could have won by more, but Welbeck's second-half header was ultimately enough to secure the points on a day the England striker found himself on the left wing due to Olivier Giroud's return from injury.
Laurent Koscielny stepped back into the Arsenal defence to add a degree of calm and control, the Frenchman largely keeping the in-form Saido Berahino at arm's length throughout.
The recent England call-up hit the bar in the closing stages, but a deserved third defeat in a row without scoring leaves Alan Irvine's side looking nervously over their shoulders as the bottom half of the Premier League bunches up.
Koscielny and Giroud returned to Arsenal's XI after respective eight-week and three-month injury absences, though a lack of goalkeeping options saw Emiliano Martinez make a first Premier League start.
The Argentinian was afforded a comfortable outing in truth, though his opposite number Ben Foster had to be sharp early on to deny Santi Cazorla and Alexis Sanchez as Arsenal dominated the opening exchanges.
Wenger was dealt a blow midway through the half as left-back Nacho Monreal was forced off and replaced by Kieran Gibbs, further adding to Arsenal's fitness woes in a week that saw Jack Wilshere ruled out for three months.
Foster did well to close the space as Welbeck was fed by Aaron Ramsey, before surrendering possession cheaply to Giroud, but the Frenchman could not hit the net from a narrow angle.
Ramsey himself went close shortly before half-time, but the sides went in at the break level despite Arsenal firing in eight shots on goal to the hosts' zero.
Arsenal continued to dominate in the second half and, though Berahino hit the side-netting, their persistence was rewarded just before the hour mark.
Cazorla broke down the left flank and past Andre Wisdom before clipping an inviting cross which Welbeck powered beyond Foster - though the keeper could perhaps have done better having got a glove to the effort.
The goal finally sparked West Brom's attack into life, but Berahino saw his looping header bounce back off the woodwork and Ramsey got a crucial toe to Craig Gardner's fierce drive.
The well-earned points move Arsenal back into top-four contention, though a midweek meeting with Southampton and subsequent trip to Stoke City will once again provide an examination.