Daniel Nava and Dustin Pedroia each knocked in a run, while A.J. Pierzynski added three hits and scored twice for the Red Sox, who were coming off three straight home losses to Milwaukee but won their first at home this season thanks to a 14-hit attack.
John Lackey (2-0) allowed a single unearned run on five hits with two walks and five strikeouts over seven solid innings.
Shin-Soo Choo and Alex Rios collected a pair of hits and Mitch Moreland drove in the lone run for the Rangers, who had dropped two of three over the weekend at Tampa Bay.
Tanner Scheppers (0-1) pitched well in defeat, giving up two runs on nine hits over five full frames.