Moore (8-7) struck out six over 5 1/3 innings, scattering six hits while walking three to earn his third win in four starts.
Tampa Bay came into the game tied with the Orioles and Detroit 1 1/2 games behind American League wild-card leaders Oakland and Los Angeles.
Desmond Jennings and B.J. Upton each socked a solo homer to provide the offense for Tampa Bay, which won the opener of a three-game series.
Baltimore starter Tommy Hunter (4-7) took the hard-luck loss, surrendering just two runs on six hits over 5 2/3 frames.
2014: The Year in Music [PHOTOS]