The two sides were locked at 1-1 following extra time at the Hawthorns, after Thomas Eisfeld's opener for the visitors had been cancelled out by Saido Berahino in normal time.
But misses from Craig Dawson and Morgan Amalfitano in the shootout allowed Nacho Monreal to smash home the decisive spot-kick to book Arsenal's place in the next round.
Steve Clarke made eight changes to the side that beat Sunderland 3-0 on Saturday, with only Youssouf Mulumbu, Sinclair and Stephane Sessegnon retaining their places.
Arsenal too took the opportunity to rest key personnel, with Per Mertesacker and Serge Gnabry the only survivors from the 3-1 win over Stoke City, while Nicklas Bendtner made his first start for the club in two years.
West Brom had the first chance in the 25th minute when Dawson's looping header from Graham Dorrans' corner required Lukasz Fabianski to tip over the crossbar.
Arsene Wenger's side were then lucky not to have 18-year-old debutant Isaac Hayden sent off. The midfielder’s sliding tackle on Mulumbu was late and robust but referee Robert Madley deemed it only a yellow card.
West Brom produced the best effort of the match on the stroke of half-time, when Reid’s 35-yard free-kick dipped and just grazed the top of the bar, but that near miss saw the teams go in level.
Mulumbu forced the first genuine save of the night on the hour mark, when he teed himself up for a volley that Fabianski parried out for a corner, but West Brom failed to orchestrate clear openings and were duly punished as Arsenal went ahead in the 62nd minute.
Bendtner’s sideways pass into the box was met by the run of Eisfeld and the 20-year-old showed good composure to coolly slot beyond Luke Daniels at the near post.
That goal should have allowed the visitors to exercise control of the game but, with 20 minutes remaining, the home fans were on their feet celebrating an equaliser.
Arsenal failed to clear their lines from a West Brom corner and a dinked cross from Dorrans was met by Berahino, who headed home from eight yards despite Fabianski getting a hand to the ball.
Both teams pushed forward for a winner in the ensuing minutes, but with neither able to find a breakthrough, the game headed for extra time.
Bendtner could have wrapped it up in the 97th minute when Serge Gnabry sent him racing clear, but the Dane took too long to get his shot away and allowed Dawson to recover and make a last-ditch tackle.
And with just four minutes remaining, substitute Amalfitano thought he had won it for West Brom but saw his drilled effort cannon back off the bar.
That meant the match headed to penalties where, after two successful spot-kicks for the hosts and one for the away side, Gnabry saw his effort saved.
But Dawson and Amalfitano both fired wide, leaving Monreal the chance to win the tie, and the Spaniard made no mistake as he fired high into the roof of the net.
