Odorizzi (2-3) fanned seven and walked just two for the Rays, who have won two straight on the heels of a 1-6 stretch.
Tampa Bay second baseman Ben Zobrist left in the fifth inning after dislocating his left thumb on an attempted steal of second.
Brandon Maurer (1-2) allowed two runs on two hits and four walks over 3 2/3 frames for Seattle, which has dropped three of its last four games.
There were just five total hits in the game.
The Rays plated their two runs in the fourth.
Dominic Leone came on for the Mauer and bounced a slider in the dirt that rolled to the backstop and Myers came home to make it 2-0.
Seattle hadn't advanced a runner past first until the eighth when Dustin Ackley smoked a one-out double into center. John Buck came up next and struck out on a controversial check-swing.
Mariners manager Lloyd McClendon stormed out of the dugout to argue with first base umpire Lance Barksdale and was ejected. He tossed his hat into the stands prior to exiting the field of play.
2014: The Year in Music [PHOTOS]