The tie appeared to be heading for defeat when Noah Ajayi gave United the lead in the first half.
Benji Casey then won and converted a penalty deep into stoppage time to level the scores, with the foul also resulting in Albert Mills being sent off for United.
Palace pressed for a winner throughout extra time, striking the woodwork on multiple occasions, before Dean Benamar converted the decisive kick in the shootout.
Under-18 head coach Javier Alonso said: "I don't have words for the boys. They did an unbelievable job this season. They gave their best, fighting under the lights of Selhurst Park with the supporters. It's an unbelievable achievement for the squad, for the boys. I'm really proud."
The result came five days after Palace were knocked out of the FA Youth Cup semi-finals by the same opponents — a Manchester United side that had won all 14 of their matches in 2026.
Alonso added: "I said to the boys today: 'we cannot finish this season without a trophy'. The mentality of the players is unbelievable. The other day we lost against Manchester United but we knew that at Selhurst Park it would be different. Before penalties I just said to the boys: 'you are top players. You deserve to be here under the lights of Selhurst Park. We are going to win. Just choose one side, go with confidence.'"