Eduardo Nunez's go-ahead RBI triple opened the floodgates in a four-run seventh inning that also featured a two-run homer from Derek Jeter, capping an emotional 24 hours after Rivera was lost for the season with a torn ACL and meniscus in his right knee.
"You've got to have a short memory here," said Jeter, who is batting .404 and leads the Majors with 44 hits. "Every time you go out, you have to forget about what's happened, and we played well tonight. CC pitched well; he deserves a lot of credit, and Nuney came up with a big hit."
David Robertson was summoned for the ninth inning in a non-save situation as the Yankees' bullpen went into operation for the first time without Rivera since 2003, the closer's last trip to the disabled list. Rafael Soriano warmed up for the eighth but did not enter.
"I wouldn't say I really thought about it, but I knew I had to get three outs and finish the game," Robertson said. "It's going to be like that, I guess, from now until the end of the season."
The Yankees gave Sabathia a first-inning lead facing Bruce Chen, as Teixeira continued his dominance against the journeyman left-hander with a two-run homer. It was Teixeira's seventh homer in just the 20th at-bat of his career against Chen.
"It's really all about that I see him well," Teixeira said. "There's certain guys that you just really see the ball out of their hand."
Sabathia quickly gave the lead back in the bottom half of the first inning, as Alex Gordon doubled home Jarrod Dyson and Jeff Francoeur doubled home the second run.
"I just didn't make pitches," Sabathia said. "We had chances to get the outs; I just didn't make the pitches. I was able to make adjustments in the dugout, go out and just try to pound the strike zone."
Sabathia was able to pin the Royals from there, including escaping with runners at third base in the second, sixth and seventh innings.
"He bore down," said catcher Chris Stewart, working for the fourth straight start with Sabathia. "He basically learned from his mistakes in that first inning and did what he did the rest of the game."
Sabathia, who allowed two runs on seven hits, retired 12 straight at one point and finished with five strikeouts and no walks. He said that he shook off Stewart only once, having done so zero times in the last two starts.
"CC steps up when we need him the most," Jeter said. "He likes to be out there. He likes to finish games. He threw eight great innings today."
Chen was able to dance away from trouble in the middle innings, but his luck ran out as the Yankees cobbled together a two-out rally in the seventh.
"Bruce made three mistakes that cost him five runs but, boy, after the first inning, he settled in and matched Sabathia pitch-for-pitch," Royals manager Ned Yost said.
Nunez slugged a well-placed triple that landed in deep right field to drive home Robinson Cano with the go-ahead run, and Stewart lined a single over shortstop Alcides Escobar for the fourth run.
Jeter completed the damage against Chen by jumping on a changeup and supplying the power, belting a two-run blast to left field, the fifth of the year for the shortstop, who hit six all of last season.
"The bottom of the order really contributed tonight with two-out hits, starting with Nuney," Yankees manager Joe Girardi said. "It just kept going. It was important, and Jeet kind of capped it off with his two-run homer to give CC some breathing room."
The lighter mood in the clubhouse was a welcome change from the deathly silent atmosphere on Thursday, as the stunned Yankees contemplated life without their 42-year-old closer.
"I don't think anybody thought he was going to retire," Sabathia said. "He's still on top of his game, still the best closer in the game, so I was hoping he wasn't thinking about retiring."
The Yankees still must figure out how to get through 2012 without his gifted right arm, but as Rivera told his teammates, no player is bigger than the team and continuing to win games is the most important thing.
"He just said, 'Don't feel sorry for me, I'm going to come back,'" Robertson said. "We're a good team, injuries happen, let's turn it around, get stronger and go from there."