Fans yawned through another laboured performance in the pouring rain as the 11-times winners failed to turn overwhelming dominance into goalmouth thrills against a third-tier side 48 rungs lower on English football's ladder.
Wayne Rooney finally broke the deadlock with a well-taken 90th-minute penalty after substitute Memphis Depay went down theatrically over the lunging leg of midfielder Dean Hammond.
It belatedly raised a cheer from the home crowd but more questions will now be raised about the future of manager Louis van Gaal who watched on impassively.
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While Van Gaal said the most important thing was to be through to the next round, former England international Scholes was not buying it.
"I've tried so hard to defend this team but each week it gets harder," he told BT Sport. "Every time I come to Old Trafford it's negative football, I would be depressed if I was part of that performance.
"The players are bored, fans are bored, everyone's bored. You look across at the bench and Van Gaal looks bored as well."
While the result may offer a little breathing space for the Dutchman, whose side recently went eight games without a win, the manner of the victory was far from a convincing endorsement of the conservative tactics that have prompted scorn from fans and former players.
It was the 10th home match in a row that United failed to score before halftime and at one point, when Depay had a speculative shot from outside the area, ironic cheers broke out.
Rooney, United's attacking talisman for so many years, had only 34 touches and while United made 672 passes, few of them were designed to open up Sheffield's stubborn rearguard.
"It's always difficult to play against such an opponent when they are defending with 10 or 11 players behind the ball," said Van Gaal.
"We had to play more quickly than we did in the first half. It was a bit better in the second, we created a bit more but not much.
"They (Sheffield) were well organised but it's more easy to defend than attack," added Van Gaal.
(Editing by Tony Jimenez)
