Díaz began his career with the Naranjas de Villa Clara in the 2008-09 Cuban National Series; at 16 years old, he registered two hits and two walks over seven plate appearances.

In 2013, at 21 years old, Diaz defected from Holguin, Cuba to Monte Cristi, Dominican Republic alongside his childhood friend Leandro Linares and another unnamed individual.

[5][6] Díaz began the 2016 season with Akron and was subsequently promoted to Columbus, where he won the International League Rookie of the Year Award.

Díaz hit two home runs in the 2019 American League Wild Card Game as the Rays defeated the Oakland Athletics.

[11] Although plagued by a hamstring injury in September 2020, Díaz appeared in 34 games of the abbreviated 2020 season, slashing .307/.428/.386 with 2 home runs and 11 RBI.

[12] In the 2022 season, Díaz hit .296/.401/.423, leading the Rays in on-base percentage (.401), OPS (.824), and OPS+ (142), and finished second only to Harold Ramírez in batting average.

In his first at-bat, he hit a solo home run to put the AL up 1-0, becoming the first Rays starter to homer in an All-Star Game and the first Cuban-born player to do so since Cookie Rojas in 1972.

Díaz was announced as part of FEPCUBE's "Patria y Vida" team of expatriate Cuban ballplayers participating in the inaugural Intercontinental Series in Barranquilla, Colombia.