Moreland added two singles for the Rangers, who won for the third time in four games to capture the series after losing the opener.
Rodriguez (2-2) allowed one run and four hits, struck out five and walked two after not getting out of the fifth inning in his previous start, a loss to Cleveland last Friday when he gave up eight hits and five runs..
Shawn Tolleson got three outs for the save after earning the first of his career on Wednesday.
Clay Buchholz (2-5) gave up three runs -- two earned -- and five hits in 7 2-3 innings.
David Ortiz drove Boston's run, tying Fred McGriff for 43rd all time with his 1,550th RBI. Boston, which came in with the AL's second-worst batting average and ranked 12th in runs scored, hasn't scored more than four runs in nine straight games.
The Rangers jumped ahead 2-0 in the first. Shin-Soo Choo and Delino DeShields opened with singles and advanced on Prince Fielder's flyball to the right-field warning track. Choo scored on Moreland's fielder's choice grounder and DeShields scored on the same play when shortstop Xander Bogaerts dropped the ball for an error attempting to take it out of his glove to complete a double play.
Moreland's homer into the first row of Green Monster seats made it 3-0 in the fourth.
Ortiz's run-scoring grounder cut it to 3-1 in the bottom of the inning. The Red Sox had runners on second and third with no outs, but didn't get a hit.
Rangers: OF Josh Hamilton joined Triple-A Round Rock to continue his rehab for at least three days.
Red Sox: RF Shane Victorino was in the original starting lineup, but had to be scratched with a sore left calf. ... Boston manager John Farrell said 3B Pablo Sandoval "was much improved today. There's still some soreness there" so they didn't start him. He pinch hit and struck out in the eighth. He was hit by a pitch on the side of the left knee on Tuesday. ... Farrell also said RHP Joe Kelly's right wrist was fine after his was hit by grounder in Wednesday's start. Kelly had it wrapped after the game.
Rangers: RHP Colby Lewis (3-2, 3.06 ERA) is scheduled to start the opener of a three-game series in New York on Friday against Yankees RHP Michael Pineda (5-1, 3.31).
Red Sox: Rick Porcello (4-2, 4.26 ERA) is slated to start Friday against Los Angeles Angels' RHP Garrett Richards (3-2, 2.29). Porcello is 3-0 with a 2.84 ERA in his last five starts.