In a match that finished goalless after extra time it fell to United defender Victor Lindelof to score the decisive spot-kick after Solly March sent his effort high over the bar.
The result means Erik ten Hag's side have reached their second domestic final of the season, having won the Carabao Cup in February.
In a scrappy semi-final that fell short of expectations, both goalkeepers excelled with David de Gea making several fine saves and Brighton's Robert Sanchez brilliantly turning away Marcus Rashford's deflected effort in extra time.
March's miss came after 12 successful penalties, before Lindelof stepped up to set up a mouth-watering all-Manchester encounter back at the national stadium on 3 June.
City beat Sheffield United 3-0 in the other semi-final on Saturday.