Christian Eriksen netted a fine free-kick in the first half to put us 1-0 up on the night and 2-0 in front on aggregate.
We held onto that lead with relative ease before a quick-fire double from Che Adams brought the Blades right back into the game with 10 minutes remaining.
We kept going, though, and Harry Kane played in Eriksen to coolly slide home our second goal in the 88th minute to give us a 3-2 aggregate victory and book a Wembley final date with Chelsea.
United: Howard, Flynn, McEveley, Basham, Harris, Doyle (Reed 65), Campbell-Ryce (Adams 73), Scougall, Murphy, McNulty (Higdon 73), Baxter. Substitutes (not used): Turner, Alcock, Kennedy, Wallace.
Spurs: Vorm, Walker, Vertonghen, Dier, Davies, Stambouli, Mason (Rose 90), Lamela (Townsend 73), Dembele (Paulinho 65), Eriksen, Kane. Substitutes (not used): Friedel, Fazio, Soldado, Adebayor.
Goals: United – Adams 77, 79; Spurs – Eriksen 28, 88.
Yellows: United – Doyle, Flynn; Spurs – Walker, Mason, Dembele.
Referee: Mike Dean.