Barton's sacrifice fly in the second inning broke a 2-2 tie and his RBI single in the seventh gave the A's an insurance run, boosting Oakland to its third win in four games.
Sheets (2-3) was effective, striking out eight en route to the victory. He yielded just two runs on four hits.
Three A's relievers held the lead over 1 2/3 innings. Andrew Bailey pitched a scoreless ninth to earn his sixth save.
The Rays dropped only their third of 16 road games this season. John Jaso and Jason Bartlett led Tampa Bay with an RBI each. Wade Davis (3-2) took the loss, allowing three runs on six hits in five innings.