Detroit had won the first two games of the series, each by one run, but weren't competitive in the finale.
Odorizzi (7-8) held Detroit to six hits and a walk while striking out six.
Kyle Lobstein (3-7) took the loss in his second start back from a shoulder injury, giving up five runs, seven hits and two walks in 5 2-3 innings.
The Tigers, who currently have a 17-man pitching staff, used 14 pitchers between Tuesday night and Wednesday, including all 12 of their relievers.
Tampa Bay broke the game open with three runs in the fourth on run-scoring doubles by Logan Forsythe and Asdrubal Cabrera.
Detroit had a big early scoring chance in the third when Anthony Gose doubled and took third on center fielder Brandon Guyer's error. With one out, the Rays brought in the infield, and Ian Kinsler hit a line drive directly at third baseman Evan Longoria.
Odorizzi walked Miguel Cabrera on four pitches, and Guyer made a leaping catch of J.D. Martinez's 400-foot drive to left-center, saving two runs.
With one out in the fourth, Mikie Mahtook walked and took second on Longoria's single. Forsythe blooped a double to right, scoring Mahtook, and Cabrera followed with a two-run double into the gap in left-center.
Detroit put two runners on in the fifth with no outs, but Gose bunted into a force at third, Kinsler flied out and Cabrera struck out to end the inning.
Forsythe's solo homer made it 4-0 in the sixth, and Tim Beckham added his eighth home run of the season two batters later to chase Lobstein. J.P. Arencibia hit Tampa Bay's third homer in two innings off Buck Farmer in the seventh, giving the Rays a 6-0 lead, and Mahtook made it four with a 430-foot blast to the flag pole in center field.
The Rays added a third run in the seventh on Beckham's RBI single.
Rays: OF Steven Souza went 3 for 5, including a three-run homer, while rehabbing with Class-A Charlotte in the Florida State League playoffs.
Tigers: DH Victor Martinez (illness) was out of the starting lineup for the third straight game, but Tigers manager Brad Ausmus said he was available off the bench. . Farmer left the game after Arencibia's homer with forearm tightness.
Rays: Tampa Bay is off Thursday before a three-game weekend series with Boston.
Tigers: Detroit goes to Cleveland to start a four-game series. Alfredo Simon (12-9, 4.86) faces Danny Salazar (12-8, 3.54) in a rematch from Saturday. In that game, Simon pitched seven shutout innings while J.D. Martinez and Tyler Collins homered off Salazar in a 6-0 win.
Wednesday marked the 38th anniversary of Alan Trammell and Lou Whitaker making their major league debuts as Detroit's double-play combination in the second game of a doubleheader with Boston. They combined for 4,683 games and 39 seasons, all with the Tigers, and were key members of Detroit's last World Series title in 1984.