Martínez was born in Bahía Blanca. He followed in his father's footsteps to become a professional footballer, joining local side Liniers, with whom he excelled at U–17 level.^[9] In 2013, he scored 13 goals in the U–17 league and netted in the final of the National Cup, though Liniers ultimately lost on penalties to Rosario.^[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.^[10] Soon after joining the club, Martínez began suffering from homesickness and wanted to return to his hometown. He was ultimately convinced by teammate Braian Mansilla to stay and went on to score 53 goals in 64 appearances for the club's reserve side.^[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. Martínez's father, Mario later revealed that he chose to stay with Racing at the time as he was not ready to leave the club.^[12]His league debut followed on 1 November 2015 when he came on as a second-half substitute for Diego Milito in a 3–0 win over Crucero del Norte.^[13] On 17 April 2016, he was sent off for the first time in his career after picking up two bookings in the space of five minutes in a 2–2 draw with Argentinos.^[9] He scored his first goal later that year, netting the opener in a 1–1 draw with Huracán in November.^[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.^[15] Reports followed that Martínez had signed for Atlético Madrid side but Racing later announced that he had signed a renewed contract which included an increased release clause. They also criticised their Spanish contemporaries for submitting him to a medical without their permission.^[15][16] He ended the year with a man of the match performance, scoring once and assisting another in a 3–1 win over Gimnasia.^[17] On 27 February, he scored his second hat-trick of the season when he scored three times on his Copa Libertadores debut against Brazilian side Cruzeiro. Racing won the match 4–2.^[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. Blanco indicated that Martínez was set to join Inter ahead of the 2018–19 Serie A campaign but that he would attempt to keep him on loan until December 2018 in order for the striker to continue to participate in the Argentine side's Copa Libertadores campaign.^[19][20]

Martínez officially joined Inter on 4 July 2018 for a reported fee of €22.7 million and he signed a five-year contract with the club.^[21] He made his unofficial debut ten days later and scored in a 3–0 friendly win over Swiss club, Lugano.^[22] His full debut followed on 19 August when he started in a 1–0 Serie A loss to Sassuolo in the opening week.^[23] He scored his first league goal against Cagliari, the opener of a 2–0 win at San Siro on 29 September.^[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.^[25] Following this match, he began to start in every match, profiting from Mauro Icardi's absence for personal reasons.^[26] In his first Derby della Madonninamatch against city rivals Milan, Martínez first assisted Matías Vecino's header before scoring himself a penalty in the second half, helping Inter to win 3–2;^[27] Inter retook the third place in Serie A,^[28] and also achieved the first league double over them since 2011–12 season.^[29]

On 2 October 2019, Lautaro Martínez scored his first goal in UEFA Champions League in a 2–1 away defeat to Barcelona in group stage;^[30] he became the first player since Roberto Boninsegnain 1970 to score at Camp Nou for Inter.^[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.^[32][33] On 17 August 2020, Martínez scored a brace in a 5–0 win against Shakhtar Donetsk to reach the 2020 UEFA Europa League Final.^[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.^[35]

On 16 May 2023, he scored the only goal in a 1–0 victory over rivals Milan in the Champions Leaguesemi-final second leg, which secured a 3–0 win on aggregate for his club and qualification to the Champions League Final for the first time since 2010.^[36] On 24 May, he scored a brace in a 2–1 victory over Fiorentina, to achieve his second consecutive Coppa Italia.^[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. Martínez began the season with an immediate impact for Inter, scoring both goals in a 2–0 home win over Monza on the first match day, and then scoring Inter's second goal the following week in another 2–0 win away against Cagliari.