After pitching two scoreless frames, Odorizzi surrendered a leadoff home run to Will Middlebrooks in the third.
Xander Bogaerts added an RBI single off Odorizzi in the fourth that tied the game at 2.
Odorizzi left the game after walking Mike Carp to start the sixth. Cesar Ramos took over and surrendered a two-run homer to Sizemore that cleared the boardwalk in right field and landed on the walkway outside the Rays' executive offices.
Odorizzi allowed three runs on four hits and two walks while striking out five in five-plus innings.
"I'm really happy [with the outing]," Odorizzi said. "It was good to get out there more relaxed this time than the past outings this spring. And I think that made a big difference going out relaxed and comfortable. I think it had a great deal to do with today and I'm really happy how today went."
Clay Buchholz started for the Red Sox and surrendered two runs in the third on an RBI double to Jose Molina and an RBI single to Wil Myers that gave the Rays a 2-1 lead.
Buchholz allowed two runs on three hits and two walks while striking out five in six innings in his last Grapefruit League start.
Up next: Left-hander David Price will take the mound for the Rays on Wednesday against the split-squad Orioles in a 7:05 p.m. ET contest in Sarasota, Fla., on MLB.TV and Gameday Audio. The outing will mark the Tampa Bay's ace final tuneup in advance of his Opening Day start Monday against the Blue Jays at Tropicana Field. Right-handers Juan Carlos Oviedo, Brandon Gomes and Mark Lowe, along with left-hander Cesar Ramos, are also expected to pitch.
Right-hander Chris Tillman will start for Baltimore.