The Rays ended a four-game losing streak, with Price getting the best of Detroit ace Justin Verlander.
Price (11-4) gave up two runs on five hits and struck out seven. He was removed from the game in the eighth inning after complaining of tightness in his lower back.
Jake McGee worked the eighth and Fernando Rodney turned in a perfect ninth for his 22nd save.
Jennings hit a solo homer in the third to give the Rays a 2-0 lead and he hit another in the fifth. Ben Zobrist hit a solo shot in the fourth off Verlander (8-5).
It was the fourth time in Verlander's eight-year career that he had given up three home runs in a game.
The defending American League MVP and Cy Young Award winner allowed six hits in six innings while striking out eight.
Detroit lost for the third time in four games. One of its runs came on a homer by Delmon Young.
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