The Seattle Mariners selected Leone in the 16th round of the 2012 Major League Baseball draft, and he received a $100,000 signing bonus when he signed with the team. The Mariners assigned him to the Short Season-AEverett AquaSox, where he made 19 appearances and posted a 3–0 record, 1.36 ERA, 39 strikeouts in 33 total innings. In 2013, Leone began the season with the Class-AClinton LumberKings. After pitching scoreless innings for Clinton, the Mariners promoted Leone to the Advanced-AHigh Desert Mavericks of the California League, and served as their closer. In July, Leone was promoted to the Double-AJackson Generals, where he ended the 2013 season. In 48 combined appearances across three minor league levels, Leone posted a 1–3 record, 2.25 ERA, and 64 strikeouts in 64 innings. In the offseason, he made 11 relief appearances for the Peoria Javelinas of the Arizona Fall League, and recorded a 3.00 ERA and 15 strikeouts. On April 4, 2014, the Mariners selected Leone's contract from the Triple-ATacoma Rainiers of the Pacific Coast League, promoting him to the major leagues. He remained with the Mariners for the entire 2014 season, making 57 relief appearances and posting an 8–2 record, 2.17 ERA, and 70 strikeouts in 66 innings. During spring training in 2015, Leone struggled with his command, and the Mariners assigned him to Triple-A Tacoma. When the Mariners placed Tom Wilhelmsen on the disabled list in early April, they promoted Leone. Leone made 10 appearances with the Mariners in 2015, posting a 6.35 ERA and nine walks in 11 innings. In the minors, he put up a 1–1 record, 7.71 ERA, and eight strikeouts in 9.
Arizona Diamondbacks
On June 3, 2015, Leone, along with Welington Castillo, Gabby Guerrero, and Jack Reinheimer, was traded to the Arizona Diamondbacks for Mark Trumbo and Vidal Nuño. Leone made three appearances for the Diamondbacks in 2015, and went 0–1 with a 14.73 ERA in 3 innings. With the Double-A Mobile BayBears, Leone made 19 appearances and pitched to a 1–2 record, 3.90 ERA, and 28 strikeouts in 27 innings. Leone was ejected by Vic Carapazza in a game against the Miami Marlins for hitting Christian Yelich with a pitch in retaliation to Jose Fernandez hitting David Peralta earlier that game. In 2016, Leone split time between the Diamondbacks and the Triple-A Reno Aces. With Arizona he posted a 0–1 record, 6.33 ERA, and 23 strikeouts in 27 total innings. With Reno, Leone went 5–2 in 33 relief appearances, and recorded a 3.34 ERA and 36 strikeouts in 35 innings. In November 2016, Leone was designated for assignment by the Diamondbacks.
Toronto Blue Jays
On November 18, 2016, Leone was claimed off waivers by the Toronto Blue Jays. Though initially sent by the Jays to the minor league camp late in spring training, Leone was recalled on April 2 to start the 2017 season with the major league club, taking the roster spot of the injured closer Roberto Osuna. He was optioned to the Triple-A Buffalo Bisons on May 30 but was recalled again on June 6. Leone was placed on the bereavement list for undisclosed reasons on August 13, and activated on August 16. On September 11, Leone earned his first major league save when he pitched a scoreless 9th inning in a 4–3 victory against the Baltimore Orioles. Leone would finish his first season as a Blue Jay appearing in 65 games, posting a 2.56 ERA, and striking out 81 batters over 70 innings. Leone qualified for super two status during the 2017-18 offseason, and signed a one-year, $1.085 million contract for the 2018 season on January 12, 2018.
St. Louis Cardinals
On January 19, 2018, the Blue Jays traded Leone and Conner Greene to the St. Louis Cardinals for outfielder Randal Grichuk. On May 9, he was placed on the 10-day disabled list with nerve damage in his right arm, and on May 30, he was transferred to the 60-day disabled list. In 13 innings pitched prior to his injury, he compiled a 4.15 ERA with 15 strikeouts. He was activated from the disabled list on August 26. Leone finished his first season in St. Louis with a 1-2 record and a 4.50 ERA in 29 relief appearances. Leone began 2019 in St. Louis' bullpen, but was sent down to the Memphis Redbirds in May after pitching to an 8.02 ERA in innings. He was recalled to St. Louis on June 25, but optioned again on July 21. He was recalled once again on August 22, finishing the season in St. Louis. Over relief innings pitched during the regular season with the Cardinals, Leone went 1-0 with a 5.53 ERA, striking out 46. Leone was designated for assignment on November 20, 2019. He was released on November 25.
Cleveland Indians
On January 23, 2020, Leone signed a minor league deal with the Cleveland Indians. The Indians selected Leone's contract on July 23, 2020.