Dean Smith's side will now face either Leeds United or Derby County at Wembley for a place in the Premier League.
Jonathan Kodjia, meanwhile, will presumably be working on his dab in preparation for the final on May 27 at Wembley.
Jedinak came on with just seconds remaining in the match and - after Jed Steer put Villa in control with great saves from Mason Holgate and Ahmed Hegazi - slotted his side's second spot-kick easily.
West Brom was without suspended star striker Dwight Gayle but got a goal from Craig Dawson in the 29th minute to draw level.
He did not, however, received any sympathy from Aston Villa substitute Jonathan Kodjia, who took the opportunity to - very kindly - point his opponent in the direction of the tunnel as he walked past.More news: ‘Stranger Things’ LEGO Is Coming to Take You to The Upside Down
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Villa beat West Brom 4-3 in a shootout after their semi-final finished 2-2 on aggregate following West Brom's 1-0 win in the second leg on Tuesday (Wednesday NZ time).
Brunt knew what it meant for his team as soon as he walked but West Brom held out valiantly against the extra man for the remaining 40 minutes (including extra time).
Goalkeeper Jed Steer was Aston Villa's hero with two penalty saves in the shootout.
"There's obviously a lot of luck involved but wow - we practise penalties, so I think I must've faced 100 a day the last few days", Steer said.
Villa missed out in the playoff final a year ago, 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.