Rodriguez beat a throw to second on Johnson's bases-load grounder that allowed Ben Zobrist to score the winning run.
"We win the game because a guy running at first, Sean, does the right thing," Tampa Bay manager Joe Maddon said. "We win because of him and his effort. Sean gets a great lead at first, a great jump and makes the play. That doesn't happen every night."
Evan Longoria opened the 10th with an infield single off Duane Below (0-2) but was forced out at second on Zobrist's grounder. After Matt Joyce struck out, Casey Kotchman was hit by a pitch and pinch-hitter Rodriguez walked to load the bases.
"It was fortunate ... Inge got rid of it real good, but Santiago didn't get there, I guess, early enough," Rodriguez said. "The ball beat me there, but he didn't."
Inge, a defensive replacement in the 10th, also briefly bobbled Zobrist's grounder, which might have cost the Tigers a double play.
"It's not obvious I could have gotten him at third," Inge said. "I was playing way back to give myself a better angle. He hit the ball to my left, so it's very cut and dried ... On a ball hit to your left, you go to second base all the time. That's the right move. That's the only play you have."
Tigers manager Jim Leyland said the final play is just part of the game.
"I thought there was an out at first or probably at third," Leyland said. "We didn't execute a couple of chances and it cost us. We couldn't come up with the big hit. We didn't do much offensively."
Joel Peralta (3-4) pitched a perfect 10th.
Longoria hit a solo homer for the Rays, who entered 0-5 against Detroit this season. Wade Davis allowed two runs and four hits in nine innings.
Detroit got a solo shot from Alex Avila, who has six homers and 16 RBIs in 20 games this month. The Tigers had a five-game winning streak end.
The Rays had two on and one out in the ninth, but Below got Johnny Damon to ground into a double play. Damon has hit into just three double plays in 479 at-bats this season.
Damon gave the Rays a 2-1 lead on his 2,700th career hit, an RBI double off Max Scherzer in the fifth. The Tampa Bay designated hitter also doubled in the first and moved past Mark Grace (511) into 45th place all-time with 513 doubles.
Avila pulled the Tigers even at 2 on his 16th homer this year during the seventh. He is 8-for-21, including four home runs, against Tampa Bay this season.
Scherzer gave up two runs and four hits in five innings. The right-hander was pulled after throwing 97 pitches.
Longoria tied it at 1 with his 22nd homer of the season in the fourth. It was his 104th career homer, tying him for third place on Tampa Bay's career list with Carl Crawford. Longoria has 61 RBIs over his last 64 games.
Austin Jackson put the Tigers up 1-0 on a third-inning sacrifice fly.
Davis struck out Miguel Cabrera with two runners on to the end the sixth. Cabrera flew out to deep center leading off the ninth.
Damon is the 64th player to reach 2,700 hits. ... Leyland said Tampa Bay's rotation is the best in the AL. "I really believe that," Leyland said. "Pretty impressive." ... Cabrera will go on the paternity leave list and miss Thursday's game but is expected to rejoin the team Friday. ... Catcher Omir Santos will be called up from Triple-A Toledo to replace Cabrera and will likely give Avila, who has started 18 straight games at catcher, a break. ... Tampa Bay center fielder B.J. Upton (strained right shoulder) struck out as a pinch-hitter for the second straight game. ... Rays bullpen coach Bobby Ramos (throat infection) is not with the team. ... Tigers right-hander Doug Fister (5-3) will pitch against Tampa Bay right-hander Jeremy Hellickson (11-8) in Thursday's series finale.