The Championship side trailed 1-0 at half-time to a JJ Melligan penalty, but came storming back with a double strike from Greg Halford and efforts from Billy Sharp and Kyle Naughton to book a home tie against Charlton Athletic in the next round.
League One side Orient, who had only won twice at home this season, started well and had a couple of penalty shouts for fouls on Adam Boyd and Sam Parkin turned down.
But it was third time lucky when Sean Thornton fired in a shot from 20 yards that struck Matthew Kilgallon on the hand and referee Iain Williamson pointed to the spot.
Up stepped Melligan, who had taken over the spot-kick duties from Boyd, and he calmly sent keeper Ian Bennett the wrong way.
It was the one real opening of a first half high on action but low on chances.
But the Championship side stepped up a gear in the second half and settled the game with a deadly three-goal burst in 11 minutes.
On 58 minutes Orient keeper Glenn Morris dropped David Cotterill's corner and Halford hooked the ball home from close range.
Four minutes later the Blades took the lead when Darius Henderson sliced open the Orient rearguard and Sharp fired home a precision effort into the corner.
Orient tried to respond and Boyd went close, but United killed the game off on 69 minutes.
Orient were cut open again after losing possession from a corner and after Morris saved well from Cotterill and Henderson, Naughton fired home.
Twelve minutes from time Orient's misery was complete when Kyle Walker's cross from the right was headed home by Halford and the League One side's dream of a cup shock, which had lasted for an hour, had turned into a nightmare.