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.
The result, which brings with it an estimated £40million windfall, also ended a 33-year absence from the top-flight for one of the most historic clubs in English football.
United had settled first and after Kyle Naughton had tested Brian Jensen with a volley from distance, Brian Howard also found the Burnley keeper in the way of his low right-foot shot after making a yard of space just outside the penalty area.
But despite being on the back foot from the kick-off, the Clarets scored against the run of play after 13 minutes.
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.
The Blades felt they should have been awarded a penalty just moment later when Howard appeared to be tripped by Graham Alexander on the edge of the box only for referee Mike Dean to wave play on.
Burnley could have been two up on the half-hour mark when Martin Paterson cut in off the right flank and curled a left-footed effort a couple of yards wide of the far post.
Steven Thompson too was inches away from doubling the advantage on the stroke of half-time when he met a chipped cross from Elliot with a towering header which looped inches wide of the upright.
Sheffield United looked full of energy from the restart, but again it was Burnley who appeared more dangerous in the final third.
Michael Duff was inches away from stabbing Thompson's glancing header from a corner in at the far post with the United defence all at sea.
Nick Montgomery then came to his side's rescue after 55 minutes when he inadvertently stopped Joey Gudjonsson from stabbing the ball over the line from close range with his back leg when a second goal was surely on the cards.
Kyle Walker was next to save the Blades when he charged back to deny Robbie Blake the opportunity to stroke the ball home from close range following a Thompson cross which took Kenny out of the game.
Walker and Naughton both had half-hearted appeals for penalties turned down as the Blades began to dominate possession without ever seriously threatening Jensen.
But their game was effectively up ten minutes from time when Jamie Ward saw red for picking up two needless bookings in quick succession.
Both were for deliberate handballs and with the departure of the substitute striker went the South Yorkshire club's own hopes of returning to the top-flight after a two-year absence.