The Clarets started as underdogs after ending the season fifth in the table compared to United's third-placed finish, but Owen Coyle's side produced an energetic and committed display to run out worthy winners of the game described as the richest in football.
Elliott started and finished a swift move when he picked up the ball in his own half and ran at the heart of the United defence.
He played in Chris McCann but when Matthew Kilgallon stopped the midfielder with a firm challenge, the ball broke to Elliott and he beat the despairing dive of Paddy Kenny with a superb clipped shot from 22 yards out.
Substitute Jamie Ward was sent off for Sheffield United with ten minutes remaining after picking up two bookings for delebrate hand-ball to aid his teams cause, and out with the departure of the striker went the Yorkshire club's own hopes of returning to the top-flight after a two-year absence.