Alcántara signed with the Los Angeles Angels of Anaheim as an international free agent in May 2012. He made his professional debut with the Dominican Summer League Angels that year. He spent 2013 with the Orem Owlz of the Rookie-level Pioneer League and 2014 with the Burlington Bees of the Class A Midwest League. In 2014, he was selected to represent the Angels at the All-Star Futures Game. Alcántara started 2015 with the Inland Empire 66ers of the Class A-Advanced California League. The Angels added him to their 40-man roster after the season.

After the 2016 season, the Angels traded Alcántara to the Detroit Tigers for Cameron Maybin. He pitched for the Erie SeaWolves of the Class AA Eastern League and the Toledo Mud Hens of the Class AAA International League. The Tigers promoted Alcántara to the major leagues on September 5, 2017, and made his major league debut that day.