Both teams made a number of changes for the contest and chances proved to be rare on a tight evening at the Hawthorns.
Thomas Eisfeld opened the scoring for the visitors before Saido Berahino equalised to force extra time.
The match went to penalties and, despite taking an early lead in the shootout, the Baggies were eventually knocked out when Nacho Monreal secured a 4-3 triumph for Arsenal.
Read below to find out how the drama unfolded in the third round clash.
ARSENAL: Fabianski, Jenkinson, Mertesacker, Vermaelen, Monreal, Hayden, Arteta, Eisfeld, Ryo, Bendtner, Gnabry
ARSENAL SUBSTITUTES: Viviano, Bellerin, Flamini, Olsson, Iwobi, Akpom, Park