Hellickson (6-3) yielded just a second-inning single to J.P. Arencibia en route to seven scoreless frames in the dominating win. The right-hander overcame four walks and struck out four, tossing 59-of-96 pitches for strikes.
Fernando Rodney finished up with his 16th save as the Rays took the opener of a three-game series.
James Loney, Wil Myers and Sam Fuld socked back-to-back-to-back solo homers in the second inning, providing all the offense the Rays would need.
Esmil Rogers (3-3) was tagged with the loss, yielding four runs on seven hits over six innings for the Blue Jays, whose club-record winning streak came to an end.
Jose Bautista drove in Toronto's only run with an eighth-inning RBI groundout.