The club’s record signing produced an outrageous flicked finish at a vital point of the second half to restore Spurs’ two-goal lead and help them clinch just their second Premier League win in seven.
Spurs, who have thrown away 10 points from winning positions this season, might have been feeling nervy, but Ndombele came to the rescue with his goal-of-the-season contender.
It helped Jose Mourinho’s side, who left Dele Alli at home amid speculation surrounding his future, get some momentum ahead of their next league game in 10 days against Liverpool.
For Sheffield United it was their 16th defeat of the campaign and suggested Tuesday’s first win of the season against Newcastle may have been a false dawn in terms of hopes of a great escape.
Indeed, they soon found themselves in a familiar losing position as Tottenham needed less than five minutes to go ahead.
Steven Bergwijn’s rasping drive from the edge of the area forced a fine save out of Aaron Ramsdale, who acrobatically tipped over.
But from the resulting corner, Ramsdale was in no-man’s land and Aurier nodded into an empty net from Son Heung-min’s delivery.
Son was almost on the scoresheet minutes later as Kane slipped him through on goal but, with Ramsdale advancing, his clipped effort hit the outside of the post.
Ramsdale had another scary moment as he clawed away Ethan Ampadu’s skewed clearance.
But they got the crucial second goal five minutes before the break as Kane produced a fine finish.
Receiving the ball from Pierre-Emile Hojbjerg, the England captain turned and drilled a low shot into the bottom corner from 20 yards for his 12th league goal of the season.
Mourinho’s side may have been fearing the worst when McGoldrick got the Blades back in the contest just before the hour.
The striker got to John Fleck’s inviting delivery first and glanced into the bottom corner.
But any hopes the Blades had of getting back into it were killed just three minutes later by Ndombele’s moment of magic.
Spurs won possession back high up the pitch, Bergwijn played the Frenchman in with a chipped ball and, as the danger seemed to have passed, Ndombele, running away from goal, sent a flick back over his shoulder with the outside of his boot which looped over Ramsdale and into the far corner.
The visitors had chances to rack up a more handsome win, but Kane, surprisingly, was in a wasteful mood as he twice failed to hit the target from inviting positions on the edge of the area.