Sabathia (15-5) is tied with Tampa Bay left-hander David Price for the most victories in the American League.
Sabathia, who is 11-2 in his past 14 starts, gave up three runs and 10 hits, while walking two and striking out three in 8 2/3 innings.
With closer Mariano Rivera unavailable after working four of the past five games, including the past two at Texas, manager Joe Girardi called Robertson to get the final out after Sabathia allowed two singles in the ninth.
Robertson got his first save of the season in three opportunities, but not before giving up a two-run double to Willie Bloomquist, who had three hits.
"That [Betemit single] added a little more pressure to the situation," Robertson said. "Kendall was all over me. I finally threw a good breaking ball in there and made him swing and miss. It's the hardest out I've had to get in a long time."
It was Robertson's second career save.
"Not too many of those come around, so I'll take it," he said. "When I come set, I'm looking at the tying run on third base. We don't get a chance to hit if I give up a base hit and they score the runs. There's a little more pressure."
Sabathia, who has a 35-10 career record in August, has a 17-10 record against the Royals, including a 10-5 record at Kauffman Stadium. He has won 15 games in his first 25 starts for the first time in his career.
"It's tough to come to life when Sabathia's on top of his game, and he was on top of his game," Royals manager Ned Yost said. "That's as good as you'll see him throw. He was probably 4 miles, 5 miles an hour faster on his fastball, and his breaking ball was sharper. He was definitely on top of his game.
"We couldn't get any big hits or score any more runs in the ninth than the ones we scored to keep the game close. We got the tying run to the plate in the eighth inning and the winning run to the plate in the ninth. We stayed right in it."
Sabathia has twice logged 19 victories, but never 20 in a season.
"I'd rather win a World Series than 20 games and a Cy Young," Sabathia said.
It was 94 degrees for the first pitch with a heat index of 106, which prompted Girardi to remove Sabathia after 110 pitches.
"With the heat, we don't want to wear this big guy down," Girardi said. "He's important to us down the stretch. We needed to get one out and Robby was able to do it. It's a tough call, but that's when I get paid to do."
Kearns' ninth home run was his first with the Yankees, who acquired him in a July 30 trade with the Cleveland Indians. Kearns homered on a full-count pitch with the bases empty in the fourth.
Curtis Granderson, who had not started the previous two games and was in a 3-for-29 skid, singled home Robinson Cano in the second with the first Yankees run. Cano had walked and stopped at second on a Kearns single.
The Yankees scored an unearned run in the seventh when Jeter reached on Betemit's error to start the inning. Nick Swisher, who had three hits, singled Jeter to second. He went to third on Teixeira's fly to deep center and scored on an Alex Rodriguez groundout.
Bruce Chen (7-6), who had won his first two August starts, left after five-plus innings, allowing three runs and eight hits and two walks, while striking out two. He was replaced by rookie right-hander Kanekoa Texeira after yielding a sixth inning leadoff single to Marcus Thames.
"I felt like I made one mistake to Austin Kearns and he got me," Chen said. "Other than that, I felt like I was mixing my pitches to all different angles. They're a good ballclub, they're hot, and they're playing good."
The Yankees have won 32 of their past 43 games against the Royals.
Yankees LHP Andy Pettitte was held out of a simulated game Thursday in Tampa, Fla., with a stiff left hip flexor tendon. He has not pitched since July 18, going on the disabled list with a left groin strain. ... Royals CF Gregor Blanco left after four innings with dehydration, heat exhaustion and dizziness. ...Teixeira returned to the Yankees' lineup after missing the previous two games for the birth of his son. ... Yankees RHP Alfredo Aceves, who is on the 60-day disabled list with a strained lower back, is scheduled to throw two innings or 35 pitches Thursday for Double-A Trenton on a rehab appearance. ... Royals 1B Billy Butler grounded into his American League-leading 25th double play in the third.