Born in Córdoba, García graduated from Instituto's youth setup. He made his first team debut on 6 June 2014, coming on as a late substitute in a 0–1 away loss against Defensa y Justicia for the Primera B Nacional championship. García scored his first professional goal on 24 May 2015, netting his side's last in a 2–0 win at All Boys. He subsequently established himself as a regular starter, scoring two goals in 2016. Instituto's coach at the time, Héctor Rivoira, praised García, saying that "we have a boy with special qualities. He's different than the rest. We just need to take it easy with him."
On 5 August 2016, Instituto announced the sale of García to La Liga side UD Las Palmas for a fee of US$ 440,000. The four-year deal was made official by the Canarians the following day. He made his top tier debut for the club on 10 September, starting in a 1–2 away loss against Sevilla FC. García scored his first goal for the Amarillos on 20 December 2016, netting the first in a 2–1 home win against SD Huesca for the season's Copa del Rey. On 10 January 2017, he made an assist and scored a goal in a 2-3 away win against Atlético Madrid in the Copa del Rey Round of 16. His first goal in the league came on 20 January 2017, netting the first in a 1–1 home draw against Deportivo de La Coruña. On 13 July 2017, García was loaned to CA Osasuna in Segunda División, for one year. The following 11 January, he moved to fellow league team AD Alcorcón, also in a temporary deal.
On 5 July 2018, García joined Superleague Greece side Aris on a one-year loan deal. He made his debut on 27 August 2018 and combined it with a goal and an assist, helping to a 3-0 away win against Lamia. On 17 September 2018, he sealed a 2-0 home win against Levadiakos. On 26 September 2018, in the dying minutes of the first half Mateo scored with an excellent finesse shot, equalizing the score in a 1-1 away win against PAOK for the group stages of the Greek Cup. On 30 September 2018, he was the MVP of a 2-0 home win against Asteras Tripoli, having won a penalty and scored the second goal. On 21 October 2018, he scored an early opener in a derby match against PAOK. This goal however wasn't enough to help his team take the three points, as PAOK managed to overturn the score. On 12 November 2018, Mateo opened the score before the first minute was completed in an eventual 2-1 away win against Apollon Smyrnis. On 31 January 2019, after some very poor performances, he scored the only goal in a 1-0 away win against Xanthi and three days later he defined another game in favour of his team, scoring in a 1-0 home win against PAS Giannina, in the battle for a Europa League spot. Mateo continued his excellent goaling streak, by scoring in an away match against Asteras Tripolis sealing an emphatic 3-0 win. On 5 May 2019, in the last matchday of the season, he scored a brace in a 7-2 home win against Xanthi. Mateo finished the season with 11 goals and 1 assist in 32 games.
On 18 August 2019, García signed a four-year contract with Red Star Belgrade, with his transfer from Las Palmas costing €1.8 million. His transfer was the second most expensive acquisition in Red Star's history at the time, only behind that of Richmond Boakye. On 21 August 2019, he scored a goal on his debut for Red Star in their 2-2 tie with BSC Young Boys in the 2019–20 UEFA Champions League play-off.
