Nathan Redmond opened the scoring in the 18th minute when he fired home from the edge of the area after a corner was only half cleared by the Blades defence.
Adam Rooney doubled Blues' advantage seven minutes before half-time.
The Blades came very close to reducing the deficit early in the second half by hitting the woodwork twice before Wade Elliott all but sealed the tie with a 20-yard strike to make it 3-0 to Blues.
Rooney doubled his tally for the afternoon with 12 minutes remaining by tapping home a cross from left-back David Murphy.