The only goal of the game came in the 85th minute from City striker Kelvin Etuhu - younger brother of former City and now Norwich midfielder Dickson.
Etuhu converted after being set up by substitute Ched Evans, although United defender Ryan Shawcroft almost equalised with a header in stoppage time.
City will now face either Newcastle or Brighton (who play March 1) as they attempt to win the trophy for the first time since 1986.
Micah Richards, who recently made his first-team debut and was City's FA Cup hero at Villa last weekend, was ruled out with a 'slight knee injury', while City also lost forward Daniel Sturridge in the second half.
Midfielder Michael Johnson captained the side in Richards' absence.