The hosts established a healthy lead inside 11 minutes through a terrific brace from Shane Long. The 26-year-old broke the deadlock with a superb solo effort, before doubling his tally with a delightful chip over Villa goalkeeper Brad Guzan.
Karim El Ahmadi pulled one back for the visitors midway through the second half, before Westwood levelled matters with his first goal for the club, meaning the last four ties between the clubs have ended all-square.
Villa were unchanged from the side that beat Cardiff City 2-0 last time out, while Goran Popov replaced Liam Ridgewell for West Brom to make his first Premier League start of the season.
The hosts went ahead through a piece of individual brilliance from Long in the third minute. The Republic of Ireland international latched onto a Chris Brunt pass from deep, took the ball around Nathan Baker with a wonderful first touch on the edge of the penalty area and side-footed beyond Brad Guzan for his second goal in as many matches.
Paul Lambert’s men almost hit back instantly as Aleksandar Tonev’s powerful strike was saved by Boaz Myhill, who got down well to his right.
Long then got his second, capitalising on a misplaced Leandro Bacuna pass to take the ball beyond Baker once again and dink a sublime finish over the helpless Guzan and into the back of the net off the underside of the crossbar.
Stephane Sessegnon could and should have made it three eight minutes later, but diverted wide of the left post from a Morgan Amalfitano cross when it appeared easier to score.
Libor Kozak clipped the crossbar at the other end in the 25th minute, receiving the ball from a Christian Benteke header before pulling off a sweet strike on the half-volley.
Long was in inspired form and charged through the Villa defence on the half-hour mark before firing his eventual effort straight at Guzan from 18 yards.
Sessegnon missed a second glaring opportunity in the 64th minute, blazing over the crossbar from eight yards out.
It was turning out to be a night to forget for the Benin international and Villa got themselves right back into the tie three minutes later when El Ahmadi got his boot on the end of a Andreas Weimann header to lift the ball over Myhill.
Westwood completed the Villa fightback 14 minutes from time, receiving the ball from a loose defensive header and firing a thunderbolt into the bottom right-hand corner.
