Villa beat West Brom 4-3 in Wednesday's (AEST) shootout after their semi-final finished 2-2 on aggregate following the latter's 1-0 win in the second leg.
Craig Dawson's first-half header brought the Baggies level in the tie, but after captain Chris Brunt received a red card with 10 minutes of regulation time remaining, Villa assumed full control.
Despite incessant pressure from the Villans, the hosts managed to hold on through extra-time and force a shootout.
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 with ease.
The Villans will now face the winner 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.
The play-off final is at Wembley Stadium on May 28.