Son Heung-min, who was back in action after his three-match ban for the tackle that left Everton midfielder Andre Gomes with a badly broken ankle was overturned, gave Spurs the lead against the run of play after an Enda Stevens error in the 58th minute.
David McGoldrick saw an equaliser ruled out after John Lundstram, who hit the post during the first half, was adjudged to be offside during the build-up.
But Chris Wilder's men impressively retained their composure and George Baldock's 78th-minute cross zipping into the far corner was a touch of fortune they richly deserved.
Tottenham's Paulo Gazzaniga was the busier of the two goalkeepers early on, holding a testing low drive from John Fleck after tipping Oliver Norwood's deflected second-minute strike over.
Harry Kane was unable to turn in Serge Aurier's 15th-minute cross at the near post but it proved a rare moment of encouragement for Spurs during an opening half hour that concluded with Lundstram thumping against the upright left footed.
United resumed on the front foot after half-time, Lys Mousset romping into the box and twisting around Davinson Sanchez before pulling his effort agonisingly wide.
Son was increasingly Spurs' lone outlet and gave visiting keeper Dean Henderson his first significant piece of work, arrowing a shot towards the bottom corner after Dele Alli's raking pass.
A more precise ball into the South Korea star's path led to the opener, but it came inadvertently from United defender Stevens, who gave Son an opportunity he was never likely to pass up.
McGoldrick thought he had an instant response but, after a VAR review lasting almost four minutes, Lundstram was shown to be offside by the barest of margins earlier in the move.
Justice was done as another methodical Blades move got wing-back Baldock in position in the right channel, his sliced delivery evading Eric Dier and Gazzaniga to edge Wilder's men up to fifth in the table.