Villa, relegated in 2016, will hope to avoid a repeat of last year when they lost to Fulham in the playoff final.
"It's no good getting to Wembley and falling at the final hurdle again, so fingers crossed," Villa midfielder Jack Grealish, who converted his penalty, told Sky Sport.
"All of us are very hungry and have good legs in us."
Villa won the first leg 2-1 at home but Craig Dawson's first-half header levelled it up for West Brom who had finished four points above Villa in the regular season.
The hosts had Chris Brunt sent off with 10 minutes of normal time remaining but keeper Sam Johnstone made a stunning reflex save to deny Albert Adomah and take the tie into extra time.
Villa laid siege to the West Brom goal in the extra 30 minutes and could have had a late penalty for handball but the 10-man hosts held firm to take it to penalties.
Visiting keeper Jed Steer put Villa in control with great saves from Mason Holgate and Ahmed Hegazi.
"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.
After Grealish made it 3-1, Adomah could have sealed it for Villa but lifted his kick over the bar.
James Morrison beat Steer to pile the pressure on but Abraham found the net via Johnstone's leg to spark celebrations amongst the away fans.
Leeds take a 1-0 lead over Derby into Wednesday's second leg at Elland Road for the right to face Villa on May 27.
The Championship playoff final is regarded as the most lucrative match in world soccer with last year's clash between Fulham and Villa worth about 170 million pounds even if, as in Fulham's case, the winner is relegated after one season in the top flight.
(Reporting by Martyn Herman; Editing by Toby Davis and Ken Ferris)
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