Sabathia tied the Rays' David Price for the American League lead with his 15th victory of the season, hurling one-run ball through eight innings before receiving assistance to get through the final frame and log his 11th win in 14 starts.
With Sabathia at 110 pitches and closer Mariano Rivera unavailable, Yankees manager Joe Girardi called on Dave Robertson, who pitched around trouble -- including a two-run Willie Bloomquist double -- to record his first save of the season and the second of his career.
"With the heat, we don't want to wear this big guy down," Girardi said of Sabathia. "He's important to us down the stretch. We needed to get one out and Robbie was able to do it for us."
Holding a three-run lead going to the ninth, Sabathia seemed to be on cruise control, trying to put the finishing touches on his longest outing of the year, assisted by a crackling two-seamer that induced double plays and erased baserunners.
"I always say when it's hotter, it's easier for me to get loose," Sabathia said. "I don't have to work as hard to throw the ball hard. That's pretty much how it's always been my whole career."
"He was probably four or five miles an hour faster on his fastball and his breaking ball was better," said Royals manager Ned Yost.
The start was something Sabathia had prepared for by chugging fluids throughout the team's flight from Dallas, pounding Gatorade and entire water bottles between each inning Thursday -- a formula that seemed to work to a T.
"He's no joke -- he's as advertised, he's good," Bloomquist said. "When he shows up, you'd better have your A-game if you want to have any chance to get him."
Alex Gordon and Mike Aviles logged one-out singles in the ninth, and after a fielder's choice, Girardi made what he called a "a tough call" and decided Sabathia was running out of steam.
"I felt good. They got a couple of hits there in the ninth," Sabathia said. "That's Joe's job to judge when I'm done. I think he did a good job."
Rivera's perpetual ease in the ninth inning seemed deceptive to Robertson, who served up Bloomquist's booming double and couldn't make a play on a Wilson Betemit tapper that trickled up the third-base line. An eight-pitch battle with Jason Kendall followed, ending with a hard-fought strikeout.
"It was a battle out there," Robertson said. "That's the hardest out I've had to get in a long time. ... We don't get another chance to hit if I give up a base hit, and they score the runner on second, too. It's a little more pressure."
In a game that saw the Royals' Gregor Blanco removed for dehydration and the Yankees' Nick Swisher lifted because he looked "exhausted" to Girardi, the Yankees got to Bruce Chen for three runs and eight hits in five-plus innings, including Austin Kearns' first home run as a Yankee in the fourth inning.
But it was Granderson's progress that may have been the most encouraging sign, as the center fielder went 2-for-3 with a walk, including a pair of hits off the left-handed Chen.
Granderson had turned to hitting coach Kevin Long on Tuesday for what Long called a "complete reformation" of Granderson's hitting mechanics, a set of tweaks that he brought into Thursday's action.
"It felt pretty good," Granderson said. "Regardless of the results, I felt like I was comfortable. I felt like I was able to see the ball, I felt like I wasn't up there contemplating too much. Those are good baby steps to get those things started.
In the second inning, New York touched the journeyman lefty Chen for a run in the second inning as Robinson Cano worked a one-out walk, moved up on a Kearns hit and scored on Granderson's RBI single.
The Yankees tacked on in the third as Derek Jeter legged out an infield single, went to third on a Swisher hit and scored on Mark Teixeira's sacrifice fly.
Kearns teed off on Chen in the fourth for a solo homer to left to make it a 3-1 game, and Alex Rodriguez added an RBI groundout in the seventh against Kanekoa Texeira, who attended Spring Training with the Yankees this year as a Rule 5 Draft selection.
It all added up to help Sabathia get to No. 15, which seems to put him in striking distance for what would be his first career 20-win season. But Sabathia promises that won't be fueling his second-half stretch drive as the dog days of August morph into September.
"That's something that I can't think about. I just want to go out and keep having good starts and helping this team win," Sabathia said. "I try to stay away from personal goals, to be honest. I think that everybody who's asked me that the last couple of years, I'd rather win a World Series than win 20 games or win a Cy Young."