The result sets up the first all-Manchester final in 40 years, with Manchester City awaiting at the Etihad Stadium. The two clubs last met in the final in 1986, when City won 3-1 on aggregate.
JJ Gabriel had given United the lead with a composed 76th-minute finish, chipping the keeper to put Darren Fletcher's side on the verge of the final. Palace levelled within a minute when keeper Cameron Byrne-Hughes fumbled a routine ball for Raihaan Anderson to tap in. Extra time followed, with Palace twice going close to winning it in regulation — winger Benji Casey lobbing over before a loose ball struck the post and was cleared off the line.
With penalties looming, United substitute James Overy split the Palace defence with a through ball that found Obi, who rounded goalkeeper Lucca Benetton and finished calmly to seal it with three minutes remaining.
Obi's goal was his 16th of the campaign. Gabriel, who limped off in extra time with a groin problem, ended the night with 23 goals this season. Samuel Lusale, who moved from Palace's academy in 2024, played a key role in creating both goals after his introduction as a substitute in the 68th minute.
United laboured past Peterborough, Derby, Oxford and Sunderland to reach the semi-final. The final must be played before 9 May.