[10][11] Martínez's form at youth level caught the attention of Racing Club interim coach Fabio Radaelli who subsequently signed him in January 2014.

[9] In 2015, a deal was concluded between Racing Club and Spanish side Real Madrid for Martínez's signature but he chose to remain in Argentina.

[9] Following an injury to regular starting striker Lisandro López, Martínez assumed a more senior role for the 2016–17 season, scoring 9 goals in 23 league appearances for the campaign.

[9][14] He continued to lead the line for Racing during the first half of the following season and in December 2017 underwent a medical with La Liga side Atlético Madrid.

[18] That same month, Martínez was linked with a transfer to Italian side Inter Milan and on 5 May Racing Club President Victor Blanco confirmed that the sale had gone through.

[24] Martínez scored his first European goal on 14 February 2019, netting a penalty to help Inter win against Rapid Wien, in the first leg of 2018–19 UEFA Europa League round of 32.

[31] Later in the competition, by scoring a brace in a 3–1 away win against Slavia Prague, Martínez reached two milestones: he became only the fourth Inter player (after Hernán Crespo in 2002, Christian Vieri in 2003 and Samuel Eto'o in 2010) and the fifth Argentinian player (after Hernan Crespo in 2002, Lionel Messi in six occasions, Sergio Agüero in 2019 and Ezequiel Lavezzi in 2013) to score in four consecutive Champions League games.

[34] On 8 March 2022, Martínez scored against Liverpool in a 1–0 victory at Anfield, marking Inter's first Champions League knockout goal in 3,600 days.

[37] Martínez began the 2023–24 season, his sixth with the club, as the first choice striker, albeit once again without his strike partner Romelu Lukaku after he returned to Chelsea finishing the end of his loan spell.

On 30 September 2023, after replacing Alexis Sánchez in the 55th minute of Inter's 4–0 win against Salernitana, Martínez became the first player in Serie A history to score 4 goals as a substitute.

[43] As a result of the injury, he started Argentina's opening match at the tournament against England from the bench and was later sent off after video technology indicated that he had lashed out at Fikayo Tomori.

[52] In Argentina's final group match against Qatar on 23 June, he scored the opening goal in a 2–0 win, which enabled them to advance to the knock-out stages of the competition.

[57] On 6 July, Martínez scored the opening goal in a 1–1 draw in the semi-finals against Colombia; he later netted his spot kick in Argentina's eventual 3–2 penalty shoot-out victory to progress to the final.

[58][59] On 1 June 2022, Martínez scored the opening goal and provided an assist for Ángel Di María in a 3–0 win over reigning European Champions Italy at Wembley Stadium in the 2022 Finalissima.

[61][62][63] On 9 December 2022, during the quarter-final, Martínez scored the winning penalty in Argentina's 4–3 shoot-out victory against the Netherlands to send his side into the semi-finals.

[68] In the final against Colombia, Martínez scored the only goal in the 112th minute of the game during extra time, putting Argentina up 1–0, and eventually winning them the tournament.

[70] A quick and agile player, he is known for his dribbling skills, his prolific goalscoring, his positioning in the box, and his physical strength and aerial prowess despite not being particularly tall;[71][72][73] however, he has been criticised by pundits for his lack of accuracy when taking penalties.