[1] When he was 11 years old, his parents sent him to live with an older sister in the capital, Santo Domingo, where he could both study for school and attend baseball practice.

[3] He made his professional debut in 2014 with the Dominican Summer League Cardinals and spent the whole season there, going 1–9 with a 3.97 ERA in 12 games (11 starts).

On December 14, 2017, the Cardinals traded Alcántara, along with Magneuris Sierra, Zac Gallen, and Daniel Castano to the Miami Marlins for Marcell Ozuna.

[12] He made his Marlins debut on June 29 as the starting pitcher, pitching five innings in which he gave up one run on three hits and five walks while striking out two, earning the win as Miami defeated the New York Mets 8–2.

[15] Alcántara was the sole Miami Marlins player named to the 2019 Major League Baseball All-Star Game.

[22] Alcántara regressed from his Cy Young campaign in 2023, making 28 starts and posting a 7–12 record and 4.14 ERA with 151 strikeouts in 184+2⁄3 innings pitched.

[25] He began a rehab assignment with the Triple–A Jacksonville Jumbo Shrimp, but was shut down for the remainder of the season on September 23 after experiencing forearm tightness.

[26] On October 6, Alcántara underwent Tommy John surgery to repair the UCL in his right elbow, ruling him out for the 2024 season.

Alcántara hosted multiple fundraising events in 2019 to collect baseball equipment for underprivileged youth in his native Dominican Republic.