The Gunners were second best at The Hawthorns but triumphed on penalities, despite being behind at one point in the shootout.
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The first half was a sloppy affair, with plenty of misplaced passes from both sides.
Steven Reid clipped the top of the crossbar with a free kick in first-half injury time, meaning that the teams went in level at the break.
Despite creating little in front of goal in the first 45, it was the Gunners who took the lead against the run of play in the 61st minute.
However, the Baggies drew level through Saido Berahino 10 minutes later, profiting off some slack marking to bury a header past Lukasz Fabianski.
The final whistle went with both teams level, which therefore brought 30 extra minutes in the Midlands.
The home side finished strongly but could not find a winner, meaning that penalties were needed to separate the two teams.
First blood went to the Baggies, as Daniels saved from Serge Gnabry, but poor misses from Dawson and Amalfitano gave Nacho Monreal the chance to put the Gunners into the fourth round, which he did with aplomb.
Arsene Wenger's men will play Chelsea at the Emirates Stadium in the fourth round.