Michael Pineda (6-2) rebounded from consecutive losses and Brian McCann hit a solo shot for the Yankees, who outscored the AL champions 23-4 in their first home sweep of Kansas City since August 2007.
"Unfortunately, I feel like I let the guys down," Young said. "I feel like I could have kept this game close, maybe been able to put some pressure on their pitcher and maybe we would have pulled it out, and I let it get away."
Coming into the series, New York had lost six in a row and 10 of 11.
The slumping Royals fell into a first-place tie with Minnesota. They have dropped four straight for the first time since Aug. 28-31, mustering only five runs during the slide.
"A lot of it was Pineda. Some of it was us right now. We're not swinging the bats. We've cooled off a little bit," manager Ned Yost said. "We had opportunities to get back into that game a couple of times. We just couldn't take advantage of it."
Kendrys Morales had an infield single off Betances. The previous 33 batters he faced were a combined 0 for 29.
"Our pitchers really showed up in this series against an offense that was swinging the bat really well," Yankees manager Joe Girardi said.
"It was the right pitch selection; I just didn't execute it," Young said. "It's just unfortunate. It was a big moment and it left the ballpark."
With his 665th home run, A-Rod ended his season-worst drought at eight games and increased his career total to 1,995 RBIs. The Yankees said that's two more than Gehrig gets credit for from the Elias Sports Bureau, baseball's official statistician.
Records get tricky when it comes to runs batted in, partly because RBIs did not become an official stat until 1920. So while baseball-reference.com lists Gehrig with 1,995 RBIs and Babe Ruth with 2,214, Elias puts Rodriguez ahead of both of them and behind only Barry Bonds (1,996) and Hank Aaron (2,297).
"Unbelievable," Moustakas said. "It's a phenomenal feat and it's incredible to be here to watch it."
Rodriguez's 11th homer of the season plus a single in the seventh left him 19 hits shy of 3,000.
"You see it all. You think about it. But right now, it's about wins," Rodriguez said. "We desperately needed these three wins against a great team. That's a team over there that hopefully we'll see in October."
Moustakas, the second batter of the game, completed a cycle of sorts against Pineda with a long home run into the raised concrete bleachers in right. Back on May 15, Moustakas had a single, double and triple off Pineda as Young won 12-1 in Kansas City.
Pineda struck out Moustakas in his next two at-bats, though — the second time with two runners in scoring position and one out. The right-hander then whiffed Lorenzo Cain with a nasty slider to end the threat.
Royals OF Alex Rios (broken left hand) was scheduled to be the DH for nine innings Wednesday night in his third rehab game with Triple-A Omaha, Yost said.
Royals: After a day off Thursday, the Royals play three interleague games at Wrigley Field. RHP Edinson Volquez, who is 7-0 with a 2.90 ERA vs. the Cubs, starts the series opener Friday against RHP Jake Arrieta. Volquez has held opponents to a .195 batting average, the second-lowest mark in the American League.
"It'll be nice to get a day off, hang out and not really think about what's going on," Moustakas said.
Yankees: LHP CC Sabathia (2-6, 5.47 ERA) starts Thursday night at Oakland in the opener of a seven-game West Coast trip.