The Championship playoff final will take place on Sunday, May 26.
Tammy Abraham - Villa's striker on loan from English Premier League club Chelsea - netted the winning penalty to send the Birmingham club into the playoffs final for the second consecutive year. "The day we knew we were in the play-offs we started working on penalties", the boyhood Villa fan, who... "All of us are very hungry".
After Grealish made it 3-1, Adomah could have sealed it for Villa but lifted his kick over the bar.
West Brom had returned from Saturday's 2-1 loss at Villa Park with work to do and found an unlikely route back into the tie when defender Dawson nodded in Holgate's long throw in the 29th minute.
Leeds hold a 1-0 first-leg lead over Frank Lampard's outfit.
Villa missed out in the playoff final previous year, losing 1-0 to Fulham and Grealish, who has been linked with a summer move to Tottenham, is keen to make amends in what could be his final game for the club.
Norwich and Sheffield United were promoted automatically after finishing as the top two in the regular season.