Hellickson (6-6) was dominant in the six-inning effort, retiring 15 of the first 16 Angels hitters en route to his second straight victory.
The second-year right-hander allowed only two hits, didn't issue a walk and struck out five while pinpointing 54 of his 86 pitches for strikes.
Matt Joyce delivered an RBI single in the sixth inning to break a scoreless tie.
Zack Greinke (0-1), making his first start for the Angels after being obtained from Milwaukee in a trade Friday, took the loss despite pitching well.
The right-handed former Cy Young Award winner yielded two runs on seven hits over seven full frames, walking one and striking out eight.