Leclerc signed with the Texas Rangers as an international free agent on November 20, 2010. Leclerc made his professional debut in 2011 with the DSL Rangers, posting a 3–1 win-loss record with a 2.36 earned run average and 27 strikeouts in innings pitched. He returned to the DSL Rangers for the 2012 season, pitching to a 3–1 record, a 1.54 ERA, and 41 strikeouts in innings. In 2013, Leclerc pitched for the Hickory Crawdads of the Class ASouth Atlantic League. He had a 3–4 record with a 3.36 ERA and 77 strikeouts in 59 innings. He spent the 2014 season with the Myrtle Beach Pelicans of the Class A-AdvancedCarolina League and finished the season with a 4–1 record with a 3.30 ERA and 79 strikeouts in innings. Leclerc was worked as a starting pitcher for the Frisco RoughRiders of the Class AATexas League in 2015. He struggled to a 6–8 record with 5.77 ERA and 98 strikeouts in 103 innings. The Rangers added Leclerc to their 40-man roster after the 2015 season. He began the 2016 season with Frisco, posting a 0–5 record with a 3.52 ERA and 28 strikeouts in 23 innings. He received a midseason promotion to the Round Rock Express of the Class AAAPacific Coast League, where he posted a 2–2 record with a 2.72 ERA and 50 strikeouts in 43 innings.
Texas Rangers
The Rangers promoted Leclerc major leagues on July 5, 2016. In all, Leclerc posted a 0–0 record with a 1.80 ERA, 15 strikeouts, and 13 walks in 15 innings of major league action. Leclerc made the Texas Rangers' 2017 Opening Day roster out of spring training. He earned his first career save April 12 vs. Los Angeles Angels, entering the game with one out and a runner on second in the 8th inning with a 6–3 lead before striking out Mike Trout and getting Albert Pujols to fly out. He appeared in ten games before going on the disabled list on May 9 with a sprained right index finger. Before going on the disabled list, Leclerc recorded 18 strikeouts in innings pitched. Leclerc finished the 2017 season posting a 2–3 record with a 3.94 ERA and 60 strikeouts in innings. In 2018, Leclerc split the month of April between the Rangers bullpen and Round Rock, returning to the major league roster for good on April 30. In August, after the Rangers traded Keone Kela to the Pittsburgh Pirates, Leclerc became the Rangers' closer. Leclerc produced his best season to date in 2018, producing a 2–3 record, 1.56 ERA, 85 strikeouts, and 12 saves in innings pitched. Leclerc was named the 2018 Rangers' Pitcher of the Year by the Dallas-Fort Worth chapter of the Baseball Writers' Association of America. On March 6, 2019, the Rangers signed Leclerc to a four-year contract extension through the 2022 season, with club options for the 2023 and 2024 seasons. The deal was worth $14.75 million guaranteed over four years, with $12.25 million possible over the two option seasons. In 2019, Leclerc went 2–4 with a 4.33 ERA, 100 strikeouts, and 14 saves over innings.