Minter played college baseball for the Texas A&M Aggies, and was selected by the Braves in the second round of the 2015 MLB draft.

[15][16][17] Minter replaced fellow reliever Arodys Vizcaíno as closer due to injury, but lost the role to teammate Luke Jackson by late April.

[18] He returned to Atlanta in June, then was optioned to Gwinnett for a second time in August as the Braves traded for Shane Greene, Chris Martin, and Mark Melancon.

[23] Minter started the 2020 season at spring training with the major league team, and was optioned to Gwinnett in March.

[24][25] In Game 5 of the 2020 NLCS, Minter made history by becoming the first MLB player ever to make their starting debut in the postseason.

[23] On July 18, 2021, Minter was optioned to the Triple-A Gwinnett Stripers following the Braves 7-5 loss to the Tampa Bay Rays.

[32] Although he was activated in early July,[33] Minter returned to the injured list in mid-August with the same issue,[34][35] and planned to undergo surgery.

On January 17, 2025, Minter signed a two-year, $22 million contract with the New York Mets including an opt-out after the 2025 season.