Rodríguez signed with the Pittsburgh Pirates for a $55,000 signing bonus in March 2009. He spent the 2009 season with the DSL Pirates, going 2–5 with a 4.60 ERA over 47 innings. He split the 2010 season between the GCL Pirates and the State College Spikes, going a combined 2–2 with a 4.21 ERA over innings. He spent the 2011 and 2012 seasons back with State College. He split the 2013 season between the West Virginia Power and the Bradenton Marauders, going a combined 9–8 with a 2.70 ERA over 140 innings. He was added to the team's 40-man roster on November 20, 2013. He spent the 2014 season with the Altoona Curve, going 6–11 with a 4.84 ERA over 134 innings.
Philadelphia Phillies
On December 10, 2014, the Pirates traded Rodríguez to the Philadelphia Phillies in exchange for Antonio Bastardo. He split the 2015 season between the Reading Fightin Phils and Lehigh Valley IronPigs, going a combined 7–10 with a 6.12 ERA over innings. He split the 2016 minor league season between the Clearwater Threshers, Reading, and LeHigh Valley, going a combined 7–0 with a 2.35 ERA over innings. He made his major league debut on September 11, 2016. He produced a 2.79 ERA over 9.2 innings for the Phillies in 2016. In 2017, he posted a 1–2 record with a 6.33 ERA over 27 innings for the Phillies.
Texas Rangers
On June 13, 2017, Rodríguez was traded to the Texas Rangers in exchange for cash considerations or a player to be named later. He spent the remainder of the 2017 season with the Round Rock Express, posting a 2–0 record with a 6.33 ERA over 27 innings.
On November 28, 2017, Rodríguez signed a minor league contract with the Baltimore Orioles. He opened the 2018 season with the Norfolk Tides, going 5–2 with a 4.56 ERA over innings. On July 19, 2018, Rodríguez opted out of his minor league contract, making him a free agent.
Chunichi Dragons
On July 25, 2018, Rodríguez signed with the Chunichi Dragons of Nippon Professional Baseball on a $270,000 contract. On September 15, 2018, Rodríguez broke the NPB velocity record for a southpaw pitcher throwing down a 159 km/h pitch against the Hiroshima Carp previously beating the 158 km/h record set by Seattle Mariners pitcher, Yusei Kikuchi in 2016. He produced a 0–3 record with a 2.30 ERA over innings in 2018. Rodríguez re-signed with the Dragons for the 2019 season. He posted a 3–4 record with a 1.64 ERA and 77 strikeouts in innings in 2019. On December 2, 2019, he become free agent.
Texas Rangers
Rodríguez signed a two-year contract, with a team option, with the Texas Rangers on December 16, 2019.
