Just when it looked like Richarlison's glancing header in the first minute of added-on time had sealed victory for Spurs, United equalised with a header of their own from Matthijs de Ligt five minutes later.
United extended their unbeaten streak in the league to five games — three wins followed by two away draws — for their best run in the one-year tenure of Ruben Amorim, who is clearly benefiting from having more time to prepare for games because of no European involvement this season.
"We need to expect that in one play anything can happen and change the mood of the stadium," Amorim said.
"Today was that. I think we control well the game, but we need to do better because the game was there to take."
Tottenham have now only won one of their six homes games so far, yet showed commendable resilience to fight back from conceding a 32nd-minute goal to Bryan Mbeumo — his fifth of the campaign.
The hosts dominated the second half and grabbed an equaliser in the 84th through substitute Mathys Tel's shot that deflected in off De Ligt.
When Richarlison glanced in Wilson Odobert's edge-of-the-area shot, Tottenham looked set for all three points.
But there was still time for De Ligt to find space at the back post at a corner to direct a header goal-ward and over the line before Tottenham goalkeeper Guglielmo Vicario clawed the ball away.
Later on Saturday, Arsenal take a six-point lead into their match at Sunderland, while Chelsea host managerless Wolves in one of three other games.
Wolves were earlier in the day given permission by second-tier Middlesbrough to speak to their manager Rob Edwards about the vacancy which follows their sacking of Vitor Pereira following a winless start to the season.
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