He began his professional career in Defensor Sporting, debuting on 8 November 2011, at age 16, in the 0–2 away win against Montevideo Wanderers, substituting Brahian Alemán.

However, on 27 July Italian side Torino (the club that held the contractual rights of the player) had announced that Silva was officially transferred to Pumas UNAM.

On 1 August 2017 UNAM stated in a press conference that they would take the proper measures and possibly enact legal action against Gastón Silva by consulting with Liga MX and Femexfut executives.

In June 2018, UNAM loses the case against Silva and is ordered to pay over one million, six hundred dollars ($USD) to Italian side Torino for a player transfer that was never fulfilled.

[11] Naturally left-footed, at Defensor Sporting he was fielded as a left full-back, his preferred role, while in the national team he is often used as a central defender.