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.