Rossi began playing football as a child at the El Queso school in Solymar. At the age of six, Rossi moved to the Solymar Uruguay Sports Center, with which he competed in the Interbalnearian League of Youth Football. He demonstrated his ability in the club and caught the attention of Peñarol, where he joined their academy at the age of 12. Néstor Gonçalves discovered him and was invited to play a friendly tournament in Alegrete with the a team, where he eventually was offered a contract. At the age of 14, Rossi was the top scorer for the team's under-14 team in the Uruguayan seven tier, where he was a top scorer in 2012 with 42 goals in 27 matches. The following year, Rossi joined the Peñarol youth team in the Uruguayan sixth division, where he scored 25 times in 17 appearances. With Penarol's U-15 team, Rossi helped the team win the Apertura Tournament and the Uruguayan U-15 Championship as well as the Fabian Lomineto Cup. In 2014, he was the top scorer of the Uruguayan Premier Division Under-16 Tournament, scoring 17 goals in the first half of the year, where Peñarol came out champions of the U-16 Clausura Tournament. On 7 April 2015, he was invited to train with the first-team squad of Peñarol for the first time, along with his teammates from the under-17 team Santiago Bueno and Federico Valverde. He was promoted by Pablo Bengoechea to the first team for the 2015–16 season in August. Diego alternated the practices with their U-18 team and Peñarol's senior team, but he played with the U-17 club and in the Third or Fourth Division for the entirety of the season. He finished 2015 helping the Peñarol's U-17 team win the Apertura U-17 Tournament, and after beating Danubio in a final in which he scored a goal. For the fourth consecutive year he was the top scorer of the championship, with 22 goals plus 1 in the final, although he played 10 games in the Fourth and Third Division.
Peñarol
On 5 January 2016, Rossi began the preseason with the Peñarol first team. He debuted with the club on 16 January 2016, in the Bandes Cup final, where he came on in the 7th minute for Gabriel Leyes in a 1–0 victory against Cerro Porteño. Rossi made his debut with the club at the age of 17 years and 317 days old. At the end of the 2015–16, Rossi was demoted to the second team following a change in managerial leadership, where Bengoechea was sacked and Jorge Orosmán da Silva took over. Following a series of poor results, da Silva called Rossi into the first team, where on 20 April 2016, he made his full debut with the club, playing against Sporting Cristal in the 2016 Copa Libertadores group stage match. He scored his first professional goal five days later in a 3–1 win against Rentistas.
Los Angeles
On 14 December2017, Rossi was signed as a young designated player for Los Angeles FC of Major League Soccer, an expansion franchise that began play in 2018. On 4 March 2018, Rossi made his MLS debut in Seattle, WA against Seattle Sounders FC. Rossi's goal in the 11th minute was the first goal in Los Angeles FC history. The goal was also the fastest goal by an expansion team in MLS history. On 8 August 2018, Rossi scored his first hat-trick in MLS during a USOC semifinal match against the Houston Dynamo, a game in which LAFC ultimately lost in penalty kicks. On 18 July 2020, Rossi became the 13th MLS player and first LAFC player to score four goals in a single match, accomplishing the feat in a 6–2 victory over rivals LA Galaxy in a MLS is Back Tournament group stage match.
Personal life
Rossi earned his U.S. green card in February 2019. This status also qualifies him as a domestic player for MLS roster purposes. His father is of Italian descent, whereas his mother is of Armenian descent. Since 2018, he has received invitation to play for Armenia national football team and it's still ongoing.
