Christian Benteke netted an injury-time penalty to hand Villa a much-needed 2-1 Premier League triumph over the Baggies in midweek, but the Belgium international striker was absent when the teams reconvened at Villa Park on Saturday.
Tim Sherwood's team lacked attacking impetus in Benteke's absence and West Brom had the better of the first half following a glaring early miss from their forward Brown Ideye.
Villa were instantly more impressive in the second period, six minutes into which their captain Fabian Delph netted.
West Brom's Claudio Yacob collected a second yellow card before former Albion loanee Scott Sinclair's 84th-minute strike sent Villa into their first semi-final since 2010.
It was an achievement soured first by Villa substitute Jack Grealish receiving a second booking for simulation having only been on the field for 15 minutes.
Then, in scenes the authorities are likely to take a dim view of, a group of home supporters entered the field of play during the closing moments before the final whistle brought an exuberant pitch invasion.
Saido Berahino scored West Brom's goal in midweek and he forced Shay Given - recalled for cup action in the Villa goal - to beat away a sixth-minute drive.
Craig Gardner was the provider on that occasion and the West Brom midfielder delivered a superb right-wing cross shortly afterwards, from which Ideye should have opened the scoring.
Given kept out a 32nd-minute header from Ideye, Craig Dawson supplying the right-flank ammunition on this occasion, before the Republic of Ireland veteran arched backwards to tip Chris Brunt's deflected long-range effort over.
The hosts failed to offer a genuine first-half test to West Brom's Boaz Myhill - starting under the same circumstances as Given - until the goalkeeper gathered Sinclair's low centre at the end of a darting dribble in stoppage time.
But Villa emerged for the second period displaying far greater intensity and were soon rewarded in the 51st minute.
Charles N'Zogbia cut in from the right, drawing West Brom defenders towards him, before picking out Delph to drill a low left-footed shot beyond Myhill.
Tony Pulis' team should have been back on level terms six minutes later, but Joleon Lescott mistimed his jump to head wide at the back post after a corner from the left looped up off Villa's Tom Cleverley.
Given had to be out sharply to thwart Berahino with an hour played, while Myhill reacted similarly when Gabriel Agbonlahor and N'Zogbia combined to send Sinclair racing into the box.
The on-loan Manchester City winger then spurned the chance to give Sherwood's men breathing space as he dragged wide after Agbonlahor's 70th-minute flick on.
Dawson did well to block from Ashley Westwood and Gardner almost put through his own net as the better chances continued to fall against an increasingly frustrated West Brom.
Yacob was harshly awarded a second caution for an 80th-minute challenge on Agbonlahor, leaving West Brom under-manned as Sinclair coolly strode forward on the break and curled home to settle matters - the late unsavoury scenes an unfortunate by-product of his goal as enthusiasm among the home supporters spilled over.
