Rodríguez signed as an international free agent with the Baltimore Orioles' organization in 2010, was traded to the Red Sox during the 2014 season, and made his MLB debut in May 2015.

[3] With Delmarva, Rodríguez pitched to a 5–7 record with a 3.70 ERA in 22 games started, allowing 103 hits and 30 walks in 107 innings, while striking out 73 batters.

Rodríguez, along with Henry Urrutia and Christian Walker, represented the Orioles at the 2013 All-Star Futures Game in July.

[7] After the conclusion of the regular season, the Orioles assigned Rodríguez to the Surprise Saguaros of the Arizona Fall League (AFL).

[22] On May 28, 2015, Rodríguez earned the win in his MLB debut, tossing 7+2⁄3 shutout innings in Boston's 5–1 victory over the Texas Rangers at Globe Life Park.

[28] Rodríguez became first pitcher since the advent of ERA as an official statistic in 1912 to have begun his major league career with three starts of at least six innings with one or zero earned runs allowed and at least seven strikeouts.

[29] In September 2020 he was joined by pitcher Dean Kremer as the only two rookies to—in their first three starts—throw at least five innings, with one or no runs, four or fewer hits, and at least six strikeouts in all three starts.

[30] Overall, with the 2015 Red Sox, Rodríguez appeared in 21 games (all starts), compiling a 10–6 record with 3.85 ERA, while striking out 98 and walking 37 in 121+2⁄3 innings pitched.

[31] He began the 2016 season on the disabled list, making his first start on May 31,[32] giving up two runs in six innings while getting a win over the Orioles.

For the 2016 Red Sox, Rodríguez appeared in 20 games (all starts), compiling a 3–7 record with 4.71 ERA, while striking out 100 and walking 40 in 107 innings pitched.

[36] In early June, he went back on the disabled list, after re-injuring his knee while warming up in the bullpen for a start in Baltimore.

[36] During the 2017 Red Sox regular season, Rodríguez appeared in 25 games (24 starts), compiling a 6–7 record with 4.19 ERA, while striking out 150 and walking 50 in 137+1⁄3 innings pitched.

He made a relief appearance in Game 2; facing two batters, he gave up a single and hit a batter—both runners later scored.

He started Game 4 of the World Series against the Los Angeles Dodgers, pitching 5+2⁄3 innings while allowing four runs on four hits and two walks with six strikeouts, receiving a no decision as Boston came from behind to win, 9–6.

In late July, Rodríguez advised that he had been diagnosed with myocarditis, a heart condition typically caused by a viral infection, such as COVID-19.

[46] On August 1, the Red Sox announced Rodríguez would not pitch in 2020 in order to recover from his ongoing health problems.

[50] Overall during the regular season, Rodríguez appeared in 32 games (31 starts) for Boston, compiling a 13–8 record with 4.74 ERA while striking out 185 batters in 157+2⁄3 innings.

[51] He then made three starts in the postseason, pitching to a 1–1 record while allowing seven runs in 12+2⁄3 innings as the Red Sox advanced to the American League Championship Series.

[58] His stint on the restrictive list was largely unexplained and attributed to personal family matters, in which he was not paid and removed from the active 40-man roster.

[59] Rodríguez returned in late July and was sent to the Triple-A Toledo Mud Hens for a rehab assignment before being reactivated from the restricted list.

After experiencing renewed shoulder discomfort during a throwing session, Rodríguez was transferred to the 60–day injured list on April 17, 2024.

[67] He made his Diamondback debut the next day against the Cleveland Guardians, allowing three runs on four hits across 5+2⁄3 innings in a 5–3 victory.

[1][70] Following the trade to the Red Sox, Rodriguez worked with Bob Kipper, Portland's pitching coach, to help him improve his changeup.