With the game tied 9-9 in the ninth, Andrus drew a base-on-balls off Clayton Mortensen (1-1) to begin the frame and went to third on Josh Hamilton's single.
Red Sox closer Alfredo Aceves came in to pitch and surrendered a long fly to Beltre that sent Andrus home with the go-ahead run for Texas, which took the rubber match of a three-game set.
Andrus finished 3-for-4 with three runs scored while Hamilton added three hits, including a two-run homer and a two-run triple, as the National League West-leading Rangers improved to 6-2 in August.
The Red Sox used four runs in the seventh -- highlighted by Will Middlebrooks' three-run homer -- to forge a 9-9 tie.
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