Paul Lambert's side looked destined for a derby defeat at the Hawthorns after Albion raced into a two-goal lead within the first 11 minutes thanks to a pair of fine goals from Shane Long.
But Villa struck twice in the space of nine minutes after the break as Karim El Ahmadi reduced the deficit before Westwood's first goal for the club ensured the two midlands teams remain level on 15 points in the Premier League table.
Villa have now gone six matches without a win against Albion, with their last success coming three years ago.
Steve Clarke's hosts will be kicking themselves for finishing with just one point after a blistering start threatened to blow Villa away.
Only three minutes had been played when Republic of Ireland striker Long showed sublime technique to control Chris Brunt's long pass with one touch before flicking the ball away from Villa's Nathan Baker and firing into the net from the edge of the penalty area.
Long produced another memorable strike eight minutes later when he pounced on a misplaced pass from Leandro Bacuna and accelerated away from Baker before clipping a chip over Villa goalkeeper Brad Guzan and in off the crossbar.
Villa were shell-shocked and Stephane Sessegnon should have added to their misery moments later, but he somehow managed to fluff his lines when presented with a simple close-range chance by Morgan Amalfitano.
The visitors finally sparked into life before half-time when Libor Kozak hit the bar from Christian Benteke's flick.
Long and Benteke both went close to scoring in the final minutes of the half, but there was more drama to come.
Villa's Andreas Weimann tested Albion goalkeeper Boaz Myhill with a fierce shot that deflected off Jonas Olsson.
And Lambert's men got one back in the 67th minute when Weimann headed into the penalty area and Morocco midfielder El Ahmadi took advantage of hesitation from Myhill to hook home on the turn.
Former Crewe midfielder Westwood had gone 41 games without a goal for Villa since his £2m move from Crewe in August 2012.
But he ended that drought in the 76th minute and salvaged a point in the process.
Gabriel Agbonlahor's cross was headed away by Goran Popov and Westwood seized possession 25 yards out, hitting a low, swerving drive which flew past Myhill's despairing dive.