Soriano smacked a one-out double in the 11th off Jamey Wright (2-2), stole third and scored the tie-breaking run on Granderson's fly ball, allowing the Yankees to avoid a three-game sweep at the hands of the Rays.
Robinson Cano went 3-for-5 with a solo homer and two RBI for New York, which used the victory to move within 3 1/2 games of an American League wild-card playoff spot.
Evan Longoria belted a solo homer and drove in two run for Tampa Bay, which came into the contest tied with Boston for the AL East lead.
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