Spurs opened the scoring in the eighth minute when Cristian Eriksen sent his corner into the mixer, Davinson Sanchez fired toward the far post and Vertonghen scored from a header just in front of Paul Dummett.
The only surprise was that, after Alli's 18th-minute strike that turned out to be the victor, there was no further scoring with both goalkeepers, Slovak Martin Dubravka and Frenchman Hugo Lloris, forced to make a string of impressive saves.
But Newcastle levelled shortly after through Joselu, ghosting in-between Spurs defenders before heading home Ritchie's wonderful ball from the right flank.
Spurs, whose starting 11 contained many players who were involved in the latter stages of the World Cup, faded and were relieved to hold on for victory following Vertonghen's crucial 85th-minute block to deny Rondon.
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Sanchez might have deepened their problems with a glancing 38th-minute header which flew across the face of goal.
And Lloris then saved well from Ayoze Perez from a tight angle.
Alli passed up a chance to extend his side's lead when he bundled Kane's mishit shot straight to Dubravka. Spurs will look pretty much identical to last season after becoming the first team in the Premier League era to not bring in a player during the summer transfer window.
The Frenchman was fortunate, however, as Kenedy's control let the Brazilian down at a crucial point, when a more assured first touch would have put the on-loan Chelsea midfielder in on goal.
Substitute Rondon nearly snatched a point five minutes from time when his shot hit Vertonghen and looped over Lloris, only for the ball to come back off the crossbar.