Price (12-4) yielded three runs overall on six hits, working around three walks while striking out eight en route to the victory.
Desmond Jennings' sacrifice moved both runners up, Matt Albers came in to pitch and intentionally walked Hideki Matsui to load the bases.
Pinch-hitter Jose Lobaton then coaxed a walk to force in the tying run, and Elliot Johnson followed with a sacrifice fly to put the Rays ahead.
Will Middlebrooks belted a two-run homer for the Red Sox.
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