It was the 25-year-old goalkeeper's debut for Spurs, having joined the North London outfit in the summer transfer window after leaving Premier League rivals Southampton.
With Hugo Lloris out, Michel Vorm had been in line for a rare league run-out, but he injured his knee in training on Saturday.
Eric Dier was also full of praise for the former Gillingham stopper, saying: "It was a great day for Paulo, I thought he did really, really well".
"I think we deserved to win the game".
"There's no point beating Real Madrid on Wednesday if you can't win at here at home on the weekend as well".
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Dwyane Wade and Derrick Rose , the team's healthy big-name additions to the backcourt, remain largely ineffective. This game marks the return of Markieff Morris to the lineup, who was recovering from sports hernia surgery.
Gazzaniga was arguably the best player on the pitch, and Spurs manager Pochettino has spoken highly of him, summing up his performance in one word - "fantastic".
Palace looked the more likely to break the deadlock at the start of the second half and should have taken the lead.
The ball ended up with Son on the edge of a crowded area and he curled a delightful shot past Julian Speroni.
Zaha had half a chance to level but his final effort was neither cross nor shot, and with 76 minutes gone Spurs made another switch as Kane departed the action and was replaced by Fernando Llorente. "We have to work very hard to limit them because we know they don't need a lot of chances to score goals".
"With any luck whatsoever we would have come away with some points from the game".