Our League One opponents prevented us from creating too many chances, with the game's decisive moment coming in the 74th minute.
Jay McEveley was penalised for handball and Andros Townsend stepped up to fire the resulting penalty past Mark Howard in the Blades' goal.
We consequently take a slender advantage into next week's second leg at Bramall Lane.
Spurs (4-2-3-1): Vorm, Walker, Vertonghen, Dier, Davies, Mason (Dembele 64), Stambouli (Paulinho 75), Townsend, Eriksen, Kane, Adebayor (Soldado 64). Substitutes (not used): Friedel, Rose, Kaboul, Lennon.
United (4-1-4-1): Howard, Flynn, McEveley, Basham, Harris, Doyle, Campbell-Ryce, Scougall (Wallace 87) Reed, Murphy, McNulty (Baxter 83). Substitutes (not used): Turner, Alcock, Higdon, Kennedy, Adams.
Goals: Spurs – Townsend (penalty) 74.
Yellows: Spurs – Adebayor, Sheff Utd - McEveley.
Referee: Neil Swarbrick.