Jose Mourinho's side was held by Fulham in midweek, but secured all three points at Bramall Lane to go fourth in the Premier League.
Serge Aurier opened the scoring with an early header and Kane added a second to put Tottenham in command at half-time.
David McGoldrick halved the deficit, but Ndombele scored one of the goals of the season to give the London club breathing space as they extended their unbeaten run to seven games and left the beleaguered Blades on just five points.
Spurs were rewarded for starting so positively when Aurier capitalised on slack defending by nodding in a whipped corner from Son Heung-Min inside five minutes.
Son ought to have doubled their lead when he dinked the ball over Aaron Ramsdale, but struck the outside of the post after being sent clear by Kane.
Kane was causing the Blades all sorts of problems and the captain put Tottenham two up with a clinical finish five minutes before the break, turning sharply as he took a pass from Pierre-Emile Hojbjerg and beat Ramsdale with a measured right-foot strike from 20 yards out.
Ramsdale denied Kane a second goal when he tipped the striker's rasping long range drive over the crossbar early in the second half.
The Blades were back in it just before the hour mark, with McGoldrick heading home John Fleck's inviting cross, but Ndombele restored Spurs' two-goal advantage with a moment of magic three minutes later.
Steven Bergwijn played a one-two with the French midfielder, picking him out with a clever chipped pass, and Ndombele found the far corner from inside the penalty area with a cheeky flicked lob running away from the goal.
Kane blazed over the bar at the end of a slick Tottenham move in stoppage time, but the victory had long been wrapped up by then.