Kieran Tierney and Bukayo Saka scored excellent first-half goals for the dominant Gunners, who got a quick-fire second-half brace from Alexandre Lacazette to wrap up a confident third successive victory after triumphs against Chelsea and Brighton over the festive period.
Another win sees Arsenal - who had made their worst start to a season since 1974 amid mounting pressure on manager Mikel Arteta - rise to 11th in the Premier League and only six points adrift of the top four.
Arsenal looked in the mood from the outset, with Hector Bellerin’s low drive forcing an early save from West Brom goalkeeper Sam Johnstone after a crucial header from Branislav Ivanovic.
Saka was the early danger man and tested Johnstone again with a tame strike before providing two teasing crosses that narrowly evaded captain Pierre-Emerick Aubameyang.
Tierney lashed over from distance on the half-volley before breaking the deadlock in fine style after Matt Phillips - wearing the captain’s armband for West Brom with usual skipper Jake Livermore suspended - had been denied by Bernd Leno.
The Scotland left-back twice beat Darnell Furlong all ends up before cutting into the box and unleashing an unstoppable curling strike with his weaker right foot that flew past Johnstone and into the back of the net.
Arsenal quickly doubled their lead as their sharp passing and fluid movement caused all manner of problems for the hosts.
A tremendous slick and incisive move involving Lacazette, Emile Smith Rowe and Saka was finished off by the latter with a simple tap-in from close range.
West Brom continued to look extremely vulnerable defensively, with Johnstone tipping over after a huge deflection off Ivanovic and Rob Holding aiming a header on target.
Allardyce’s side tried to mount a response, but beyond a couple of crosses that were well-judged by Leno they struggled to make an impact.
Indeed, it could have been 3-0 at half-time after Lacazette breezed past Ivanovic all too easily near the byline, but his powerful effort hit Johnstone at the near post and bounced clear.
West Brom thought they had made the perfect start to the second period, with Matheus Pereira rifling through the legs of Leno and against the post before Callum Robinson powered in the rebound from distance.
However, the offside flag had already rightly been raised against Pereira.
West Brom’s strong start after the break quickly faded as Arsenal returned to the ascendancy, with Lacazette forced wide by Johnstone after being played in by Dani Ceballos.
The third goal came shortly after the hour mark, with Lacazette - starting in place of the rested Gabriel Martinelli - arrowing in after Semi Ajayi had turned a Saka cross onto his own post and Smith Rowe’s attempt was blocked.
The Frenchman had a second just minutes later, this time finding himself completely unmarked to knee in a Tierney cross from close range after the latter had been picked out by Aubameyang.
Arsenal - who had already lost Bellerin to a knock - subbed off Saka after a heavy challenge from Conor Gallagher that earned the Chelsea loanee a yellow card.
A mishit Willian volley was turned over by Aubameyang, who also shot straight at Johnstone soon after half-time substitute Charlie Austin had smashed well wide at the other end.
Furlong’s crucial touch denied Tierney a second from Lacazette’s cross, while Joe Willock was unable to direct Willian’s delivery on target.
West Brom had a couple of late sights of goal with the game long since over as a contest, though Leno was equal to a long-range shot from Gallagher and also got down well to push Rekeem Harper’s effort around the post and preserve a deserved second straight clean sheet.
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