Albion looked the better side for the majority of the second half, but Chris Brunt was dismissed for a couple of reckless challenges and from that point on Villa were the team asking all of the questions.
Sam Johnstone produced a superb stop late in normal time to keep his side's campaign alive, but following a goalless period of extra time it was the Villans who prevailed on pens.
Neither side was able to really take control of the match in a sloppy opening quarter of the match, but it was the Baggies who made the breakthrough just short of the half-hour mark.
Holgate's long throw from the right was met by Dawson, who got between Tammy Abraham and Ahmed Elmohamady to guide the ball into the far corner of the net, where there was nobody waiting on the line to keep it out.
El Ghazi's 20-yard shot 43 minutes in was the only other attempt in the first half, though there was a big talking point just before half time as Brunt appeared to stamp on the arm of John McGinn, an incident which went unnoticed by the officials.
Villa struggled for the majority of the second period, constantly giving away possession to invite pressure onto themselves.
Jacob Murphy's shot was blocked on the line by Mings and Rodriguez was unable to adjust his body to convert the rebound from a good position.
Momentum soon shifted Villa's way as El Ghazi forced Johnstone into action, shortly before Brunt was shown a second yellow card for lunging in on McGinn.
Villa looked the more likely to snatch the win at the end of normal time, and they would have done exactly that if not for an incredible stop from their former loan keeper Johnstone, who got down to his right to keep out substitute Albert Adomah's shot from nine yards.
The Villans were also in complete control of the ball in extra time, but all they could muster in front of goal was a tame Jack Grealish shot and a header from Keinan Davis, both of which Johnstone was equal to.
Johnstone's opposite number Steer was the hero in the shootout as he kept out Holgate and Ahmed Hegazi from the opening two Albion pens and, while Adomah was unable to take his chance to wrap up the tie, Abraham made no mistake as he beat Johnstone to set up a showdown with either Leeds United or Derby County on May 27.
WEST BROM (3-4-3): Johnstone; Dawson, Bartley, Hegazi; Holgate, Brunt, Johansen (Morrison 71'), Gibbs; Phillips (Harper 75'), Rodriguez (Leko 93'), Murphy (Adarabioyo 82')
ASTON VILLA (4-3-3): Steer; Elmohamady (Davis 114'), Tuanzebe (Jedinak 122'), Mings, Taylor; McGinn, Hourihane, Grealish; Green (Adomah 75'), Abraham, El Ghazi (Kodjia 101')