Erik ten Hag's side played out a hard-fought 0-0 draw with Brighton that went to extra-time at Wembley.
Solly March was the only Brighton player to miss in the shoot-out, with Victor Lindelof netting United's winner from the spot.
It will be United's first FA Cup final appearance since losing to Chelsea in the 2018 showpiece.
United will take on City, who beat Sheffield United 3-0 in Saturday's other semi-final, in the final on June 3.
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