The Rays had tied the game 3-3 on an Evan Longoria RBI in the fifth, then forged ahead in the next inning.
That brought up Fuld, who lined an 0-1 pitch to right field for a solid hit, sending home Escobar with the game's key run.
Wil Myers pounded a two-run homer for the second-place Rays, who took the rubber match of a three-game interleague series to remain one game behind American League East-leading Boston.
Alex Torres (4-0) tossed 1 1/3 scoreless relief innings to gain the decision and Fernando Rodney secured the win with his 27th save.
Brandon Crawford drove in a pair of runs for the Giants.