The Final will take place at Wembley Stadium on Monday 27 May (3pm kick off) and the victor will secure a place in the Premier League next season.
Ticket information for the Sky Bet Championship Play-Off Final will be announced across dcfc.co.uk in due course.
West Brom hung on throughout 30 minutes of extra time, but Steer's twin saves in the shoot-out meant Adomah's subsequent miss counted for little.
Dean Smith's side could not capitalise on that however, and as chances came and went for the Villans in extra time, the inevitable penalty shootout came about. Steer was the hero though, saving penalties from Mason Holgate and Ahmed Hegazi, as Tammy Abraham scored the winning penalty.
Steer saved his first two chances and Villa was almost ideal from the spot to move on the final, where a place in the Premier League will be on the line against either Leeds or Derby.
Leeds take a 1-0 lead over Derby into yesterday's second leg at Elland Road for the right to face Villa on May 27.
The Villans will now face the victor of Thursday's semi-final between Leeds United and Derby County, with Dean Smith's men aiming to return to the Premier League after a three-year absence.
Norwich City and Sheffield United were promoted automatically after finishing as the top two in the regular season.