Just when it looked like Spurs might register their first draw this season, Son fired an 83rd-minute shot from outside the area which somehow crept under goalkeeper Martin Dubravka to put them four points behind Liverpool who play on Monday.
It was Son's ninth goal in his past 10 appearances for Tottenham and a cruel blow for Newcastle who looked to have done enough to earn a point before the keeper's howler.
Spurs, who have now gone a record 29 Premier League games without a draw, missed several chances, with Lucas Moura heading wide from six yards, Erik Lamela heading on to the post and Christian Eriksen's flick hooked off the line by Fabian Schar.
But Mauricio Pochettino's side had twice grabbed late winners in their previous league games with Fulham and Watford, against whom Son also scored after returning from Asian Cup duty with South Korea last Saturday.
"The two games were very important. With six points we have had a great week," Son said. "It was a difficult game, but we are happy to get the six points and we deserved this one."
Newcastle, with record signing Miguel Almiron ineligible, had their moments on the breakaway with Salomon Rondon also heading on to a post. But in the end it was a familiar outcome for the Geordies, who have not won at Wembley since 1955.
Pochettino was delighted by the character of his side, who have bounced back from the disappointment of behind knocked out of both domestic cup competitions in a week.
But he questioned why they are being asked to play their next league game against Leicester on Sunday in the week of their Champions League last 16 first-leg game at home to Borussia Dortmund, who play their fixture on the Saturday.
"Twenty four hours can make a massive difference," said Pochettino. "It is difficult to understand."
(Reporting by Neil Robinson; Editing by Ken Ferris)
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