Hellickson (1-0) celebrated his 25th birthday with a dominating performance against the Yankees, yielding only three hits while walking four and striking out four.
Fernando Rodney got his second save after recording the final out for the Rays, who began the season at 3-0 for the second time in franchise history.
New York's Phil Hughes (0-1) absorbed the loss, surrendering a pair of runs on five hits over 4 2/3 innings, walking two and fanning five.
The Yankees opened a season at 0-3 for the first time since 1998.
Swim Week Miami Beach 2014 [PHOTOS]