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 Tottenham's only real moment of attacking quality was enough to seal it, Son slamming home his 20th Premier League goal.
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 - took on each other.
Palace were left to rue Gazzaniga's heroics and an open-goal miss from talisman Wilfried Zaha. They remain 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 Korea Republic international 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.
