Hellickson (3-0) notched his third win in his third major league start. He held the Orioles to three hits, walking one and striking out five.
The Rays won the rubber match of a three-game series and pulled to within one game of the New York Yankees atop the American League Central.
Crawford socked a solo homer in the first inning and added an RBI single in the third to put the Rays up for good at 2-1.
Baltimore starter Jake Arrieta (4-4) was saddled with the loss, allowing three runs on six hits over six innings.