Spurs found the going tough in heavy rain against Roy Hodgson's men and lost Kieran Trippier to injury with just 22 minutes on the clock, the full-back limping off the Selhurst Park pitch.
A 66th-minute goal by 20-year-old centre back Juan Foyth was enough to earn Spurs a ninth win in 12 league games this season.
Southampton's under-pressure manager, Mark Hughes, looked set for a first Premier League win in more than two months thanks to Manolo Gabbiadini's goal in the 20th minute, which came after Danny Ings dispossessed the dawdling Roberto Pereyra.
It's no secret Palace thrive with Zaha and struggle without him, but fans will certainly be left wondering what could've been had they have had their star man available for selection against a fairly inexperienced Tottenham back line. They have been unfortunate on a few occasions - you should see their expected goals, mate - and have picked up only eight points in 11 games.
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Spurs continue their impressive start.
Paterson hit the crossbar with his follow-up before Ivory Coast defender Samba slammed home the rebound to seal Cardiff's first win in three games.
Kane has faced questions about his form and fitness this season, amid suggestions he was looking jaded after the World Cup, but after winning the Golden Boot in Russian Federation, he has scored 10 goals in 15 appearances for Spurs this term, and Pochettino believes the changeable perception and coverage of Kane is a cultural phenomenon.
Subs not used: Guaita, Ward, Schlupp, Puncheon, Riedewald. Subs: Aurier 6, Son 6, Winks 5. They have lost nearly all their big games and have been nowhere near their seductive best, yet they are still in the Champions League, just about, and a win today would take them to within two points of the leaders Manchester City.
However, it was a much more positive experience for the young centre-back on Saturday evening, scoring the only goal of the game to win the match for his side.
Palace, on the other hand, were without the services of talisman Wilfried Zaha, who missed out through injury.