With the FA Cup fourth-round replay heading for penalties after a goalless 119 minutes, the 26-year-old striker headed in Harry Maguire's knockdown to seal the Blades' progress.
Tuesday night's tie at a wet and windy Craven Cottage look destined for penalties until the 26-year-old striker stooped to head the Blades into the next round, increasing the pressure on Fulham boss Rene Meulensteen who opted to make eight changes for the clash.
United's prize is a home tie against either Preston North End or Nottingham Forest in the fifth round, but serious questions will be asked of Fulham and its Dutch boss after such an abject performance.
With the former Manchester United assistant having made wholesale changes to the side that lost 3-0 at home to Southampton on Saturday, it was perhaps unsurprising that the visitors made the brighter start when Jamie Murphy fired wide from 20 yards.
Although the home side started to see more of the ball, the best Fulham could offer in terms of a threat was a Pajtim Kasami header that sailed harmlessly wide.
Some lovely build-up play involving Scott Parker, Kasami and Hugo Rodallega gave Clint Dempsey a sight of goal, but his effort was a wild swipe that ended a promising team move in disappointing fashion.
Sheffield United is languishing in the League One drop zone but went close to defying its stature after 35 minutes when Conor Coady's inch-perfect cross found Murphy, and he flashed his header just wide.
But it was the Premier League outfit which had the best chance of the half in injury time when Dan Burn headed Kasami's corner wide from six yards out.
Nigel Clough's United side resumed confidently after the restart with Bob Harris curling a free kick just wide of Stockdale's goal before shooting straight at the goalkeeper moments later.
The game badly needed a goal and almost got one through calamitous circumstances after 65 minutes when Damien Duff's cross was sliced towards his own goal by Neil Collins, but the defender was bailed out by Mark Howard, who made an excellent one-handed save.
Both sides looked short of confidence as extra time was required. John Brayford almost made a telling contribution when he was given space to drive through the Fulham defense, but his fierce low effort produced a superb save by David Stockdale, who clawed the ball away.
Kasami went close twice late on with a crisp shot that went wide and again when Howard tipped wide.
But United won an otherwise unremarkable contest at the death through Miller, who stooped to head in from close range after Harry Maguire had headed back across goal to send the traveling fans wild and book the Blades' place in the next round.