David Ortiz, starting his final series in the Bronx, went 0 for 5 with two strikeouts and stranded seven runners. He ended the game by fanning with two on against Tyler Clippard.
Having clinched at least a wild-card berth last weekend, Boston is five games ahead of second-place Toronto with five games left, needing just one victory or a Blue Jays loss to ensure the division championship.
Price (17-9) again struggled against the Yankees, who have only a remote chance of reaching the postseason. They closed within four games of Baltimore for the second AL wild card with five remaining.
Sanchez hit a two-run homer in the first inning, his 20th since the Yankees brought him up from the minors on Aug. 3. Playing in his 51st big league game, Sanchez matched Wally Berger in 1930 as the fastest to 20 homers.
Gregorius hit a solo drive in the sixth, also his 20th. After the Red Sox rallied, Austin broke a 4-all tie with a drive to right – all four of the rookie’s homers have gone to the opposite field.
Blake Parker (1-0) got two outs for his first big league win in two years, retiring Ortiz on a groundout with two on to end the seventh. Clippard got the save, and the Yankees (81-76) ensured they will not finish with their first losing season since 1992.
Price gave up six runs and 12 hits – matching his season high – in 6 1/3 innings. He is 1-3 with a 7.89 ERA against the Yankees this year, allowing 26 runs in 29 2/3 innings.
The left-hander had won eight straight decisions overall.
Yankees rookie Luis Cessa allowed two runs and five hits in six innings.
After Sanchez’s homer and Jacoby Ellsbury’s fifth-inning RBI single built a 3-0 lead, Boston closed in the sixth on Dustin Pedroia’s run-scoring single and Mookie Betts’ RBI. Aaron Hill led off the seventh with his first career pinch-hit homer, and Pedroia tied the score with a single against Tommy Layne.
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