The Argentine only found out on Saturday that he would make his first start for the club in place of the injured Hugo Lloris and Michel Vorm – his first Premier League start since December 2015, during his time at Southampton.
The 25-year-old overcame a rocky opening to deliver a fine performance, making three excellent saves to keep the Eagles at bay.
"Every time you play for the first time at a club it's very important and emotional too," he told BT Sport .
"[I was] nervous just in the beginning, like the first 30 seconds, and then after that you get the atmosphere and team-mates help you so it's fantastic from them also.
"Last night I saw Michel wasn't good so I prepared and the goalkeeper coach told me I would play so I was ready."
Gazzaniga's manager, Mauricio Pochettino, was not surprised by his summer recruit's fine performance, telling BT: "I think he was fantastic but it's not a surprise.
"We had him in Southampton and he played with us in the Premier League. When there was the possibility to sign here, there was no doubt about his quality.
"We have three [goalkeepers] plus younger guys with a lot of potential, so we have very good [options to] protect our goal."
Spurs' only real moment of attacking quality was enough to seal all three points, Son Heung-min slamming home his 20th Premier League goal to become the top-scoring Asian player in the competition's history, surpassing his South Korean compatriot Park Ji-sung.
Speaking to BT Sport, Son felt that his side should have won by a more handsome scoreline.
"They were a difficult opponent and not easy to play, it's always a big game, every team is difficult but it was a deserved win and we should have scored more goals," he explained.