Hellickson (8-8) allowed only one run on six hits and a walk while striking out six for the American League wild card-leading Rays.
The right-hander held an opponent to one run for the second straight outing as Tampa Bay opened a three-game set with a victory, taking their 12th win in 14 games overall.
Desmond Jennings tripled and scored the go-ahead run on a wild pitch in the third inning and later added a run-scoring single. Jose Lobaton and Elliot Johnson each supplied an RBI for Tampa Bay.
Jeff Keppinger provided a solo homer for the winners.
The loss was hung on Kansas City's Will Smith (4-5), who yielded four runs on eight hits in a 3 2/3-inning effort.
Alex Gordon drove in the Royals' only run.
Celebrity Families of 2014 [PHOTOS]