Dele Alli's absence from the teamsheet will have perturbed Spurs fans, but Gazzaniga's presence was most eye-catching, the former Southampton goalkeeper making his Spurs debut in place of the injured Hugo Lloris and Michel Vorm.
Gazzaniga had not played in the Premier League since December 2015 and overcame a rocky start to deliver a fine performance, making three excellent saves to stop Palace opening the scoring.
And Spurs' only real moment of attacking quality was enough to seal it, Son 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.
Such a scrappy affair was some comedown for Mauricio Pochettino's men after beating Real Madrid at the national stadium in midweek, but a victory was key on a day in which the four teams joining them in the top five - Manchester City, Arsenal, Chelsea and Manchester United - take on each other.
Palace were left to rue Gazzaniga's heroics and an open-goal miss from talisman Wilfried Zaha. They remain rooted to the bottom of the league, four points away from the nearest team.
There was little sign of the highlights to come when Gazzaniga was perhaps lucky not to be penalised in the third minute as he cuffed Mamadou Sakho around the head, rather than play the ball, under a teasing Zaha cross.
Palace stuck gamely to their task, repelling numerous crosses and limiting Spurs to long-rangers, and home frustrations soon turned to fears when Harry Kane went down clutching his ankle after taking a kick from Tim Fosu-Mensah.
The stand-in skipper was fine to continue, but remained a peripheral figure in the match, heading tamely over under a half-cleared cross that would not sit for a volley.
Gazzaniga will have taken a confidence boost from denying Scott Dann's powerful downward header 10 minutes before the break, the Argentine flinging out a left hand to excellently deny the visitors and he was again sturdy to deny Andros Townsend's low shot after Serge Aurier gave possession away three minutes into the second half.
Aurier was again sloppy on the ball soon after, presenting it to Townsend, who freed Zaha in on Gazzaniga - the keeper misjudged his rush and the Palace star went around him, but scuffed his finish wide.
Palace continued to knock at the door, Gazzaniga again forced to save by Luka Milivojevic after the hosts failed to clear a corner.
And the Argentine was finally backed up in the 64th minute as Spurs poured forward, Palace's stretched backline clearing twice but a third fell to Son on the edge of the area and the South Korean blasted past Julian Speroni left-footed.
Son ought to have brought up his 21st league goal for Spurs, dragging past the near post after deceiving Speroni one-on-one, and Gazzaniga had time for one final piece of assured play, nicking the ball off Ruben Loftus-Cheek to quell Palace's late attempts at a comeback.
Key Opta Facts:
- After failing to win in their first three Premier League games at Wembley, Spurs have now won their last three in a row.
- Crystal Palace have failed to score in eight consecutive Premier League away games – only West Ham have had a longer away goalless run in the competition (9 in January 2007).
- Indeed, Palace are only the second team to fail to score in their first six away games of a Premier League season, after Derby County in 2007-08 (8).
- Son Heung-Min’s goal was his 20th in the Premier League, making him the highest ever Asian goalscorer in the competition, surpassing Park Ji-Sung’s record of 19.
- Paulo Gazzaniga is the 15th goalkeeper to play for Spurs in the Premier League, but only the fourth not to concede a goal on his debut (Ian Walker, Hugo Lloris and Michel Vorm).
- Four points from their opening 11 games marks Crystal Palace’s joint-worst start to a top-flight campaign, though they finished in 11th the other time it happened in 2013-14.