The defeat left the Baggies in 19th place with 26 points, 10 points behind fourth-bottom Burnley with only three games left to play. It is the first time manager Sam Allardyce has suffered relegation from the Premier League during his 17 campaigns.
West Brom started brightly but struggled to move the ball into good attacking positions, and when they did they failed to get their efforts on target. Arsenal, on the other hand, put on a brilliant display of finishing to condemn them to the drop.
Emile Smith Rowe connected perfectly with Bukayo Saka's cross in the 29th minute to deftly volley Arsenal's opener, and Nicolas Pepe cut in from the right before curling home an unstoppable left-foot shot six minutes later.
However, Matheus Pereira pulled one back for the visitors in the 67th after a surging run from deep inside his own half ended with him scoring from West Brom's only shot on target.
That was all the Baggies could muster and Willian's brilliant 90th-minute free kick wrapped up the points for Arsenal, who are in ninth place on 52 from 35 games.
"The best way to do it is with the right attitude. The three goals were very different but really good individual and collective actions, Arsenal manager Mikel Arteta said. "We have done so many right steps, some you can see and some you can't. The foundations are there."
"Massively disappointed, [but] it's time to focus on these next three games, get as much momentum as possible to make sure we come straight back up next season," he told BT Sport.
"I am really proud of the way the boys kept going. It would have been so easy to fold out there and lose four or five nil, but everyone kept going."
West Brom manager Sam Allardyce refused to be drawn on whether he would remain at the club to try to bring them back up from the Championship next season.
"That's too early to even consider or talk about that for me, it's about getting over the disappointment and when the time is right to talk about what is going to happen next season," he said.
"In my mind, tonight is just about saying to the lads you gave it all, get back, have a day off and get ready for Liverpool at the weekend."
Third-bottom Fulham host Burnley on Monday knowing a defeat would also send them straight back down to the Championship.