Conrad's double in the second inning put the Rays ahead 3-0 and his two-run hit in the sixth handed Tampa Bay a 5-2 advantage.
Sean Rodriguez, Jose Lobaton and pitcher Brandon Gomes added RBI for Tampa Bay, which took the rubber match of the three-game series after also winning Sunday's Game 1 by a 3-2 margin.
Gomes (2-2), the third of three Tampa Bay pitchers, earned the victory with 2 2/3 scoreless innings.
Philadelphia's Cliff Lee (0-4) remained winless in 12 starts this season, surrendering five runs on six hits over seven innings. The left-hander struck out nine and walked three.
Placido Polanco went 2-for-3 with two RBI for the Phillies.
Celebrity Families of 2014 [PHOTOS]