Smith was selected by the Cincinnati Reds in the 21st round of the 2010 Major League Baseball draft out of Lipscomb University. He rose through the Reds' minor league system, reaching Triple-A in 2014. In 2015, Smith was called up to the majors for the first time on April 13, but did not make his MLB debut until June 23, allowing four runs in three innings of a no decision against the Pittsburgh Pirates. He made nine appearances with the Reds in 2015, compiling an 0–4 record with 6.89 ERA. In 2016, Smith made 32 appearances with the Reds, pitching to a 3–3 record with a 4.68 ERA. He also spent time in the minor leagues during 2015 and 2016.
Oakland Athletics
On November 4, 2016, Smith was claimed off waivers by the Oakland Athletics. He began the season with the Triple-A Nashville Sounds, until the Athletics purchased his contract on May 3, adding him to their active roster. With the A's during 2017, Smith made 26 relief appearances, recording a 4.89 ERA and a record of 2–1. He was outrighted to Triple-A on November 5, 2017, and then elected to become a free agent. ;Seattle Mariners On March 15, 2018, Smith signed a minor league contract with the Seattle Mariners. He made four appearances with the Tacoma Rainiers of the Pacific Coast League, recording a 6.10 ERA, and was released on April 21, 2018.
On May 16, 2018, Smith signed a minor league contract with the Boston Red Sox. He joined the Triple-A Pawtucket Red Sox in June, appearing in 18 games with a 5–6 record and 4.14 ERA. Smith started the 2019 season with Pawtucket. He was added to Boston's active roster on April 26 to make a spot-start against the Tampa Bay Rays; however, the game was rained out. Smith made his Red Sox debut on May 3, pitching a scoreless inning against the Chicago White Sox. He next made a spot start against the Baltimore Orioles on May 6, taking the loss while allowing four runs in innings. Smith was returned to Pawtucket on May 20, when David Price was activated from the injured list. Smith was optioned between Pawtucket and Boston several times during June; he recorded his first MLB save on June 13, against the Texas Rangers. After spending all of July with Pawtucket, Smith was recalled to Boston on August 2, when Heath Hembree was placed on the injured list. Smith was optioned back to Pawtucket two days later, and recalled to Boston on August 20. Overall with the 2019 Red Sox, Smith appeared in 18 games, compiling an 0–3 record with one save, along with a 5.81 ERA and 29 strikeouts in 31 innings. Smith was removed from Boston's 40-man roster in October and sent outright to Pawtucket. On October 21, he elected to become a free agent.
Miami Marlins
On December 18, 2019, Smith signed a minor league contract with the Miami Marlins. ;Note Smith is often referred to online and in print as "Josh A. Smith", to differentiate him from Josh D. Smith, especially since the two players were teammates for the Pawtucket Red Sox in 2018.
