The Gunners were without the services of in-form Mesut Ozil through injury. His place was taken by Nigeria international Alex Iwobi.
The Baggies got off to a lively start and nearly went ahead inside six minutes but Petr Cech was alert to Jay Rodriguez's header.
The Gunners had to wait until the 17th minute for the first chance when Alexis Sanchez played in Iwobi, who controlled well before placing his effort just over the crossbar.
In the 37th minute, Arsene Wenger was forced into a first-half change as Ainsley Maitland-Niles replaced Sead Kolisanc, who was forced off with an injury.
The final stages of the first half failed to provide any goal-mouth action and the sides went into the break goalless.
After a quiet start to the second half, the Gunners nearly broke the deadlock in the 60th minute but Ben Foster produced a fine low save to deny Alexandre Lacazette from close range.
With 15 minutes remaining, Wenger introduced Danny Welbeck for Iwobi as they pushed for a winning goal, while West Brom brought on Allan Nyom for Hal Robson Kanu.
The Gunners broke the deadlock in the 82nd minute when Sanchez's deflected free-kick sailed into the back of the net.
However, West Brom were awarded a penalty in the 88th minute when Callum Chambers was adjudged to have handled in the box. Rodriguez stepped up and converted the spot-kick, 1-1.
Both sides pushed for a winning goal in the closing stages but they were unable to carve out another and were forced to settle for a share of the spoils.