Oakland surrendered four runs in the first inning but came back with five in the third.
After Yoenis Cespedes and Jonny Gomes (two RBI) each delivered run-scoring singles, Inge followed by pulling a 3-1 pitch deep into the left field stands, giving Oakland a lead it wouldn't relinquish.
The third baseman finished with four RBI and Barton's double in the fifth put the A's up 8-4.
Tommy Milone (4-2) earned the victory despite allowing five runs on five hits over five frames as the A's avoided a three-game sweep at Tropicana Field.
Rays starter Matt Moore (1-2) was pounded for eight runs on seven hits before leaving after 4 2/3 innings.