Solly March fired his spot-kick over the crossbar before Victor Lindelof converted to win it for United.
It finished 0-0 after extra time and Brighton had chances to win it but goalkeeper David de Gea denied Alexis Mac Allister, Julio Enciso and March.
Marcus Rashford nearly won it for United late on but Seagulls goalkeeper Robert Sanchez made a superb save.
Manchester City beat Sheffield United 3-0 in the other semi-final on Saturday and the final will take place at Wembley on Saturday, 3 June.
Meanwhile, Newcastle United's humiliated Tottenham Hotspur with five goals in 21 minutes to run out easy 6-1 winners in their English premier league football match.
The game was billed as a showdown for a top-four spot and the result is a significant blow to Spurs' dwindling hopes of securing Champions League qualification.
In contrast the Magpies are now in a commanding position to finish in the top four as they climb to third.
Newcastle's opening goal came within just 62 minutes of the opening whistle, while only Manchester City have scored five goals in the league in less time, doing so in 18 minutes against Watford in 2019.
And West Ham have taken a big step towards Premier League safety by thrashing fellow strugglers Bournemouth 4-nil.