Looking to become the seventh and youngest member of the 600-homer club, Rodriguez, representing the potential tying run, ended an 0-for-4 night by grounding into a fielder's choice facing Rafael Perez.
It was the biggest moment of the night for the Yankees' lineup, which was held almost completely in check by rookie right-hander Josh Tomlin, who dominated New York in his Major League debut.
The 25-year-old Tomlin faced the minimum through six innings and limited the Yankees to a run on three hits in seven-plus frames, including retiring the first nine batters he faced.
"My first pitch in the big leagues was to Derek Jeter. That's pretty cool," Tomlin said. "I was just trying to throw strikes and get them to put it in play; let them do what they do. Luckily, it panned out."
Robinson Cano doubled to chase Tomlin in the eighth, sending him off to a standing ovation from the crowd of 27,416. Pinch-hitter Colin Curtis knocked in New York's only run with a groundout, further exacerbating an oddity that has seen the Yankees struggle against pitchers they have not yet seen.
"We've become robots after a while," Rodriguez said. "We've seen guys so many times, over and over again, that over the last five or six years for some reason, these guys have had a lot of success against us. Give this young man credit; he threw the ball really well."
The Yankees threatened in the ninth against Perez, as Brett Gardner singled and Jeter punched a hit down the right-field line. That set up Rodriguez with a chance to homer for the first time since clubbing No. 599 off the Royals' Robinson Tejeda last Thursday.
Rodriguez had slogged through another quiet night, grounding to shortstop in the second inning, bouncing to first base in the fifth and flying out to right field in the seventh on a ball that might have gone out at Yankee Stadium.
But in Cleveland, it settled harmlessly into Shin-Soo Choo's glove well shy of the wall, setting up Rodriguez for his last chance in the ninth inning. Yankees manager Joe Girardi admitted to dreaming big in that spot.
"This would be a real good time to hit one," Girardi said he was thinking.
It wasn't meant to be, as Rodriguez chopped the ball to shortstop Asdrubal Cabrera for a routine force play. New York starter CC Sabathia was saddled with his first loss against his former team, accepting a defeat for the first time since May 23.
Sabathia had been 9-0 with a 2.22 ERA in his last 10 starts, but the Indians got to him just enough, nicking the ace for four runs (two earned) in seven innings, scattering nine hits among a season-high 123 pitches.
"I felt like it was so-so," Sabathia said. "I had good stuff. I just didn't make the pitches I needed to. I felt like I had a good fastball and the changeup was better than the last time. It just didn't work out [Tuesday]."
The Indians took the lead against Sabathia in the fourth inning, as catcher Francisco Cervelli lost the ball on a tag play at the plate that scored Cabrera with the first Cleveland run.
"He slid right through his mitt, so there was really nothing we could do about it," Sabathia said.
Sabathia then appeared to induce an inning-ending double play ball on Jhonny Peralta, but second-base umpire Jerry Meals ruled that Cano was off the bag. Peralta was also called safe at first base, and instead of being out of the inning, the Yankees now faced a bases-loaded situation with only one out.
"That's a call that's not made very often. It happens a lot, but it's not made very often," Girardi said.
Girardi briefly argued the call of the neighborhood play, tossing his hands up in frustration in apparent displeasure over the call at first base more than the second-base ruling. Matt LaPorta lifted a sacrifice fly to right field that gave Cleveland a 2-0 lead.
Austin Kearns opened the Cleveland sixth with a triple to center that hit Curtis Granderson's glove near the wall, later scoring on LaPorta's double to left-center.
After a mound visit and an intentional walk to load the bases, Sabathia lost No. 9 hitter Chris Gimenez to a full-count offering that forced in the fourth Indians run. That proved to be more than enough support for Tomlin, who walked none and struck out two.
Rodriguez had been 6-for-13 with two doubles and four RBIs since belting No. 599 upon arrival in Cleveland, but he has gone hitless in eight at-bats at Progressive Field.
He admits that he might be pressing due to the ever-present flashbulbs and changes to specially-marked hologram baseballs, but hopes that the drive to right-center off Tomlin in the seventh might be the indicator of big events to come.
"I think for me, the biggest thing is to stay within the game and not try to do too much," Rodriguez said. "I think a few times over the last three or four days, I've kind of come out of it and tried to swing a little too hard, or maybe get a little pull-happy.
"As long as I swing at strikes and think [about] the big part of the field, like that ball to right-center, then good things are going to happen."