Within two months of his MLB debut, Díaz broke the Mariners' record for consecutive strikeouts and was named the team's closer.

[1] While growing up in Puerto Rico, Díaz played as an outfielder during his youth along players like Carlos Correa and José Berríos.

[6] He spent the season with the Clinton LumberKings and was co-selected the organization's minor league Starting Pitcher of the Year along Jordan Pries.

[12] He recorded the first save of his career the following day, in a 5–4 win over the Boston Red Sox, in which he struck out the side.

Díaz's 24 saves when entering a game with a one-run lead broke the record for most in a season, which had been 23, set by Francisco Rodríguez for the Angels of Anaheim in 2008.

[21][22] On August 24, Díaz notched his 49th save of the season, setting a Mariners single-season record when he closed out a 6–3 win over the Arizona Diamondbacks.

[23] On August 25, 2018, Díaz became the youngest pitcher to collect 50 saves in a season when he closed a 4–3 victory over the Arizona Diamondbacks.

On December 3, 2018, the Mariners traded Díaz, Robinson Canó, and $20 million to the New York Mets for Jay Bruce, Jarred Kelenic, Anthony Swarzak, Gerson Bautista, and Justin Dunn.

Writing for Deadspin, David Roth described Díaz's downturn as "arguably the most dramatic and most surprising" of any player's collapse to that point in the 2019 MLB season.

In the pandemic-shortened 2020 season, Díaz converted six of ten save opportunities and recorded a 1.75 ERA with 50 strikeouts and 14 walks in 25+2⁄3 innings.

[32] Díaz closed out the ninth inning of a 1–0 win versus Philadelphia on August 13 to earn his 200th career save, the sixth active player to do so.

[38] On March 15, 2023, after striking out the side in the ninth inning against the Dominican Republic, sending Puerto Rico to the World Baseball Classic quarterfinals, Díaz suffered an injury to his right knee during the celebration and had to be helped off the field.

On March 16, the Mets announced that he would undergo surgery to repair a full-thickness tear of his patellar tendon and would miss roughly 8 months in recovery.

[39] On September 18, the Mets announced that Díaz would not pitch for the team in 2023, citing not wanting to jeopardize his health to rush him back during a likely losing season.

[45] He was activated from the IL and returned to the closer role on June 13 against the Marlins and earned the win that day for the Mets.

[48] On September 30, in the penultimate game of the season, Díaz threw a season-high 40 pitches across the eighth and ninth innings against the Atlanta Braves, securing the win and clinching the 11th playoff berth in Mets history.

[50] On October 3, Díaz recorded his first postseason win after throwing two scoreless innings in Game 3 of the NL Wild Card Series against the Milwaukee Brewers.

[53] Díaz played as a relief pitcher, more specifically as a closer for the Puerto Rican national team in the 2017 World Baseball Classic (WBC) where he won a silver medal.

However, it did not qualify as an official perfect game per the Elias Sports Bureau, due to lasting fewer than nine innings.

[56] Díaz also closed the final pool play game against the Dominican Republic, sending Puerto Rico to the quarterfinals.

[60] In the offseason, Díaz spends time in his native Daguao in the Naguabo Municipality in Puerto Rico, where most of his family lives.