The 22-year-old Rodriguez was already the youngest Red Sox pitcher to make his MLB debut on the road since 21-year-old Roger Clemens in 1985. Billy Rohr was 21 when he won in his debut at the New York Yankees on April 14, 1967.
Rodriguez (1-0) struck out seven and walked two, leaving the game with two runners on base in the eighth before Tommy Layne got Shin-Soo Choo on a called third strike.
Boston went ahead to stay when Mookie Betts had an RBI single in the fifth off Nick Martinez (4-1). Ramirez, who was the designated hitter with slumping slugger David Ortiz getting a night off, led off the sixth with his 11th homer of the season to make it 2-0.
Hamilton was 2-for-4, lining a double into the right-field corner on the first pitch he saw from Rodriguez in the second inning. He added an RBI single in the ninth, on another sharply hit ball to right, to end Boston's shutout bid.
Back in Texas a month and a day after being re-acquired from the Los Angeles Angels, the 2010 AL MVP had gone 1-for-11 in his first three games after reuniting with the team Monday in Cleveland.
Martinez struck out five and walked two while scattering nine hits over seven innings. The righty allowed two runs.
Boston added three runs in the eighth, including Blake Swihart's two-run single after two walks and a hit batsman the previous three batters.
Red Sox manager John Farrell indicated before the game that Ortiz could get several days off, giving him a chance to work on some things. Ortiz had one hit in 20 at-bats his previous five games, and is hitting .216 with six home runs and 18 RBIs in 43 games this season.
Red Sox: SS Xander Bogaerts sustained a bruised left forearm after being hit by a pitch near his wrist in the second inning. The pitch ricocheted off his forearm and high into the screen behind home plate.
Rangers: LHP Derek Holland threw for the first time since April 10, the home opener when he threw just one inning before leaving with strained left shoulder. Holland said he made 20 throws from 60 feet and "felt good." ... LHP Matt Harrison (spinal fusion surgery) is scheduled to throw a bullpen session Sunday in Texas. Harrison struck out nine with no walks in five innings of an extended spring training game Thursday in Arizona.
Red Sox: Boston RHP Steven Wright (2-1, 3.68 ERA) faces Texas for the first time in his career.
Rangers: Texas righty Yovani Gallardo (4-6, 4.13) has allowed three earned runs or less in eight of his last nine starts.