Bouzat started his senior career in 2011 with Liniers, playing twelve times in Torneo Argentino B.[4][5] He joined Boca Juniors of the Argentine Primera División in early 2012.[4] In January 2016, Bouzat completed a season-long loan move to fellow Primera División team Defensa y Justicia.[6] He scored on his professional debut, in a 2–2 draw with Unión Santa Fe on 7 February.[1] In total, Bouzat went on to make forty-five appearances for the club and scored three goals.[1] He returned to Boca Juniors ahead of the 2017–18 campaign and made his league debut on 23 September 2017 in a 0–4 win against Vélez Sarsfield.[1]
Ahead of January 2018, Bouzat departed Boca Juniors to join Vélez Sarsfield.[7][8] His first appearance for Vélez came versus former club Defensa y Justicia on 27 January.[1]
In June 2022, he was loaned to Colo-Colo by Vélez Sarsfield on a deal for eighteen months with an option to buy.[9][10]