Gazzaniga was born in Murphy, Santa Fe, the same small town in Argentina where his future Southampton and Tottenham manager Mauricio Pochettino was also born. His father is Daniel Gazzaniga, a footballer, as is his brother Gianfranco who plays as a goalkeeper for SD Ponferradina. Gazzaniga studied in the only public school and a secondary college in Murphy. He played for the football team of Centro Recreativo Unión y Cultura in Murphy as a goalkeeper.
In 2007, when he was 15, Gazzaniga left his hometown side Unión y Cultura with his brother and father for Spain, where he joined the Spanish club Valencia's youth setup. However, he failed to make a first team appearance for Los Che, and was subsequently released in May 2011.
In July 2011, Gazzaniga signed a two-year contract with League Two club Gillingham, to challenge Ross Flitney for the number one spot. Gazzaniga was recommended to Gillingham manager Andy Hessenthaler by Gary Penrice, Wigan Athletic's European scout, following his release from Valencia. He made his professional debut for the club on 4 October, in a 3–1 home defeat to Barnet in the Football League Trophy.
On 20 July 2012, Gazzaniga signed a four-year deal with newly promoted Premier League side Southampton, with Tommy Forecast going to Gillingham on loan as part of the deal. He described it as "a crazy situation" but said it was like "a dream".
Gazzaniga made his debut on 28 August in a 4–1 victory against Stevenage in the second round of the League Cup. He made his league debut almost a month later, on 22 September in a 4–1 win against Aston Villa.
He made two Premier League appearances in the 2015–16 season, in the 1–0 defeat away to Crystal Palace and a home 2–0 defeat to Tottenham Hotspur, both in December 2015. He signed a new four-year deal with Southampton on 11 September 2015.
On 23 August 2017, Gazzaniga signed for fellow Premier League club Tottenham Hotspur on a five-year contract. He reunited with his compatriot and former Southampton manager, Mauricio Pochettino. He made his Premier League debut for the club on 5 November 2017, playing in a 1–0 victory against Crystal Palace. Pochettino said that Gazzaniga performed "fantastically well" on his debut.
In 2018–19 after starting for Tottenham in the Premier League against Brighton & Hove Albion and Huddersfield Town, in the EFL Cup against Watford and West Ham United and in the UEFA Champions League home fixture against PSV, he earned his first Argentina national team call-up. He became effectively the number two goalkeeper for Tottenham this season, having started in more games than Michel Vorm when Hugo Lloris was not available.
In the 2019–20 season, following Lloris' elbow injury in the opening ten minutes of a loss at Brighton & Hove Albion on 5 October 2019, Gazzaniga enjoyed an extended run as Spurs' first-choice goalkeeper.
Gazzaniga made his first international appearance on 20 November 2018, coming off the bench to replace Gerónimo Rulli in the 59th minute in a friendly match against Mexico, and helped keep a clean sheet in a 2–0 win for Argentina.
- As of match played 1 March 2020
- As of match played 20 November 2018