Cabrera left the game in the fifth inning with a left knee strain, bringing Franklin in at shortstop.
''He left a pitch up over the middle of the plate and I was able to take advantage of his mistake,'' said Franklin, who has only 77 major league at-bats and two homers this season. ''I don't know what happened and it's unfortunate that (Cabrera) had to come out of the game. I was just trying to make the most of it.''
Rodriguez hit his 32nd homer this season and 686th of his career off Jake Odorizzi (8-8) in the first hours after the team learned he is playing with a bruised left knee. Rodriguez moved past Craig Biggio into sole possession of 21st place all-time with his 3,061st hit.
Odorizzi, winning for only the second time since July 28, found his fastball working early and went with it exclusively until Bird gave the Yankees a 3-2 lead on a two-run homer in the fourth. Odorizzi needed Franklin's homer to turn the deficit into a victory.
''He came up with the big hit and got us the win,'' Odorizzi said. ''It's nice to be the beneficiary of it, but he came up with the big hit when we needed it and that's been kind of tough to find this year.''
Odorizzi allowed three runs and three hits in six innings. Brandon Gomes pitched the ninth for his first save.
Tampa Bay center fielder Kevin Kiermaier took away a potential game-tying extra-base hit from Gardner with a running catch on the warning track to end the seventh.
''I almost feel like he knows where the ball's going to be hit,'' Yankees manager Joe Girardi said of Kiermaier. ''You watch him go get balls ... he's as good as I've ever seen.''
Rodriguez nearly had his second opposite-field homer of the game in the eighth, but his leadoff drive was caught on the warning track.
Adam Warren gave up two runs and six hits over four innings in a spot start for the Yankees.
By recording the first save of his career, Gomes became the eighth Ray to have a save this season. ''It's cool,'' the right-hander said. ''It's kind of a checkmark on the list of things you'd like to get done as a reliever.''
Yankees: Girardi said RHP Nathan Eovaldi remains shut down from throwing.
Rays: X-rays on Cabrera were negative. ... Reliever Jake McGee (left knee) could throw a simulated game Thursday.
Boxing legend Jake LaMotta watched part of batting practice on the field. LaMotta, 93, also took part in a program at the Ted Williams Museum, which is located at Tropicana Field.
Yankees: Rookie RHP Luis Severino (3-3) looks to bounce back Wednesday night after allowing a career-high six earned runs over 2 1-3 innings in Friday's 11-5 loss to Toronto.
Rays: RHP Chris Archer (12-11), Wednesday night's starter, is 5-1 against the Yankees. The lone loss came in his last start against them on Sept 6.