Each hitter provided a run-scoring double in the third inning, putting the Red Sox ahead for good at 2-1, while combining to go 5-for-9 with six RBI.
Dustin Pedroia went 3-for-4 with an RBI, while Jacoby Ellsbury added three hits, including a run-scoring two-bagger, as the Red Sox claimed the first of a three-game series.
Cook (3-5) mostly frustrated the Rangers for seven innings, surrendering six hits and three walks but letting just the one runner to reach home. He struck out two.
Rangers starter Yu Darvish (11-8) was roughed up for a third straight start, during which he has been tagged for 19 runs over 18 innings. The Japanese hurler gave up six runs on 11 hits through 6 2/3 frames in Monday's loss.
Adrian Beltre homered for American League West-leading Texas.