Joyce put the Rays ahead to stay with a two-run shot in the first inning and added a two-run double in the next frame, highlighting a six-run Tampa Bay uprising that put the contest away.
Jose Molina knocked in three runs, and Desmond Jennings went 3-for-4 with an RBI and three runs scored as Tampa Bay took the opener of a three-game series.
Jeremy Hellickson (7-7) registered the victory, holding the Twins to three runs on seven hits through five frames in the Rays' fourth straight triumph.
Minnesota starter Cole De Vries (2-3) lasted just 1 1/3 innings after getting lit up for eight runs (seven earned) on seven hits.
Celebrity Families of 2014 [PHOTOS]