Rodriguez came up next and blasted a 1-1 pitch into deep left-center field to put the Rays ahead.
Evan Longoria also homered for Tampa Bay, which remained one game behind the New York Yankees atop the American League East.
Randy Choate (3-3) pitched a scoreless seventh to pick up the victory and Rafael Soriano notched his 36th save of the season for the Rays.
Daric Barton went 3-for-3 with a pair of runs scored and Mark Ellis got two RBI for the A's.
Notable deaths of 2014 [PHOTOS]
UPI Sports Results for September 19, 2014