The heart of New York's order -- Rodriguez, Robinson Cano and Mark Teixeira -- combined to go 1-for-11 in the loss, including hitless in four at-bats with runners in scoring position.
"We've just got to keep swinging," Cano said. "We've been hitting with men on base, just right at people. There's nothing you can do about it."
The Royals did plenty of what the Yankees need to, as Alex Gordon tied a career high with four hits and Billy Butler drove in three runs, taking advantage of an ineffective start by right-hander Hiroki Kuroda.
Kuroda was chased in the fifth inning and needed some good fortune to escape with only three runs (two earned) pinned to his ledger.
"My role as a starter is to go as deep as I can into the game, and at least I wanted to go seven," Kuroda said through a translator. "But I couldn't do that, so it was really frustrating."
Kuroda was touched for a pair of first-inning runs after Derek Jeter booted a ground ball to open the game. Butler cashed a two-run double and added another run-scoring double in the fifth off Kuroda.
"We had some walks and made Kuroda work," Kansas City's Jeff Francoeur said. "We made him throw a lot of pitches (91), and that was our game plan coming in."
Kuroda has had difficulty settling into games this year. Nine of his 18 runs allowed have come in the first inning.
"I don't think I had many [first-inning] problems in the past," Kuroda said. "I think I'm average. You try to be too careful in the first, early in the game, and I think that's what happened."
The damage against Kuroda could have been worse, but he was able to induce Humberto Quintero to fly out with the bases loaded in the third inning. New York's bullpen assisted in the fifth, as Clay Rapada and Cody Eppley stranded two aboard.
New York was unable to manage a hit until the fifth inning against Paulino, who held the Bombers in check until Raul Ibanez broke up his no-hit bid with a liner to left field, the first ball the Yankees had been able to hit out of the infield all evening.
Paulino danced out of trouble in that inning to pin the bases loaded and keep the Yankees off the board, and three Kansas City relievers finished the job after Paulino hurled six scoreless innings.
"He threw the ball extremely well, but you know it's really about us," Rodriguez said. "We feel when we have a good approach and we take our 'A' swings, regardless of who we're facing, we've got to have better results than that."
Yankees manager Joe Girardi seemed to flash some frustration after the game, saying there isn't much he can try to change the results.
"Do you have any ideas?" Girardi said. "The guys have to go out and do it. These are our players, so they have to go out and do it. You can try shaking up the lineup, but what's that going to do? They still have to hit and be productive. I believe in these guys. They'll get it done."
Curtis Granderson ran New York out of an opportunity in the sixth, when he attempted to tag up and was cut down at third base by Francoeur's one-hop strike from right field.
"I told him, 'You've got a guy that's as good as anyone throwing out there in right field,'" Girardi said. "He got behind the ball. You can't be thrown out."
Kansas City broke the game open with two runs in the sixth, as Eppley permitted the first two batters to reach. Dyson greeted Boone Logan with a sacrifice fly, and Gordon laced a long RBI double to center, his fourth hit of the game.
Russell Martin belted a long solo home run in the seventh off Jose Mijares to break up the shutout, but the backstop's third home run of the year desperately could have used some company in an area that has shown spotty attendance.
"Nobody wants to hear excuses," Rodriguez said. "You want to hear damage, production and winning, and that's what New York's all about. That's what we expect and that's what we're going to do."