Albion - whose first XI featured eight changes and their opponents' nine - had the best chances of the first half, with a Craig Dawson header being pushed over by Lukasz Fabianski.
There was also Steven Reid's free-kick, seemingly sent over by a slight touch of Fabianski's glove, although the goalkeeper argued he had not made contact, hoping to prevent a corner that was given nonetheless.
There was no question that Fabianski had produced a save, and a good one, to keep out Youssouf Mulumbu's strike on the hour mark.
And a minute later Arsenal had the lead as Mulumbu lost possession, Bendtner slipped in Thomas Eisfeld and the 20-year-old German midfielder slotted in what was his first Gunners goal on his maiden start.
West Brom equalised in the 71st minute when Saido Berahino, scorer of a hat-trick in round two, headed past Fabianski from close range.
Baggies substitute Morgan Amalfitano hit the bar in extra time but the teams could not be separated over 120 minutes.
In the shoot-out, West Brom initially had the advantage after Luke Daniels saved Serge Gnabry's effort, but misses from Dawson and Amalfitano allowed Nacho Monreal to seal victory for the away side.
Arsenal, who have now won eight successive matches in all competitions, will host Chelsea in round four.
Source: PA
Source: PA