Myers connected for a bases-empty clout in the second inning, giving the Rays a brief 1-0 lead, then capped a go-ahead, four-run outburst in the fifth with a two-run double.
Evan Longoria went 3-for-4 with an RBI for the Rays, who took sole possession of the first AL wild card spot with the win while Texas temporarily fell into a tie with Cleveland for the second spot.
Alex Cobb (9-3) struck out 10 over eight innings, allowing two runs on six hits with a walk as Tampa Bay took the opener of a crucial four-game set.
Matt Garza (3-5) was roughed up for six runs on eight hits in a 4 1/3-inning effort for the Rangers, who dropped a seventh straight.
2014: The Year in Music [PHOTOS]