Rodriguez got a two-out RBI in the first when he was hit by a pitch with the bases loaded, and hit a three-run homer during the third that put the Rays up 4-0. He was back in the lineup after missing Thursday night's 7-2 win over Toronto for the birth of his son.
Odorizzi (1-0) allowed three hits in six innings in his seventh big league start and 10th game overall. Ryan Hanigan hit a three-run homer during the fifth for the Rays.
Rangers left-hander Joe Saunders (0-1) left in the fourth with a bruised left ankle after getting hit by Evan Longoria's liner. Saunders gave up five runs and six hits in 3 2-3 innings.
Texas slugger Prince Fielder got his first RBI since Sept. 22, a drought of 20 regular and postseason games, with a run-scoring single in the eighth.
The Rangers wasted a great scoring opportunity in the fourth. With runners on first-and-second and one out, Mitch Moreland singled, but Fielder got tagged out in a rundown between third and home. The inning ended when Donnie Murphy struck out.
Rays right fielder and 2013 AL Rookie of the Year Wil Myers had a fourth-inning RBI single that made it 5-0. He sat out Thursday's game due to flulike symptoms.
"It came on during batting practice. It was real quick," Myers said. "It went away last night. I feel better today."
Tampa Bay manager Joe Maddon won his challenge in the seventh after it was first ruled that Myers didn't make the catch while sliding forward on J.P. Arencibia's two-out flare. After a 1:43 delay, the umpires changed the call to an out.
It was Maddon's first challenge.
Notes: Texas RHP Nick Martinez will make his major league debut in Saturday night's game against the Rays. "Go out there and relax, and use his pitches," Rangers manager Ron Washington said. "Get us as many outs as you can. Be out there as long as you can. Believing in himself, believing in the stuff he has, and using it." ... LHP David Price will start Saturday for the Rays. ... Rangers LHP Matt Harrison (back) will make his next minor league start Tuesday. He went two innings in a rain-shortened outing Thursday for Double-A Frisco. ... Tampa Bay sent RHP Juan Carlos Oviedo (elbow ligament replacement surgery) to Triple-A Durham on a rehab assignment.