Caminero gave the Rays a 3-1 lead within the third on his fifth homer this season, launching an opposite-field, two-run shot to proper off former Rays pitcher Ryan Yarbrough.
Cabrera led off the fifth together with his first homer of the season and Cody Bellinger hit right into a run-scoring forceout.
Bradley gave up six runs and 9 hits in 5 1/3 innings.
Yankees starter Will Warren lasted simply 1 2/3 innings and left with the bases loaded within the second after throwing 53 pitches.
The Related Press contributed to this report.
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