The 39-year-old Kuroda gave up three hits and no earned runs over 6 2-3 innings. Cueto threw 112 pitches in five innings, matching his shortest start of the season.
"I felt really strong today," he said through a translator. "I would have kept going even with the amount of pitches I had."
Frazier won a matchup of All-Stars when he lofted Betances' full-count pitch into the left-field corner, just beyond the 318-foot sign.
It was the first home run at Yankee Stadium for Frazier, who grew up about 85 miles away in Toms River, New Jersey. He pumped his fists at home plate and received hearty high-fives in the dugout.
Betances has allowed two homers in 58 1-3 innings this season.
Moments earlier, Schumaker made a costly baserunning mistake.
He singled to start the inning and, with Frazier at the plate, took off trying to steal second before Betances began his delivery. The big right-hander stepped off the rubber, and Schumaker was tagged out after a long rundown.
Schumaker, just off the seven-day concussion disabled list, had an RBI double with two outs in the fifth. That was Cincinnati's only hit in 22 at-bats with runners in scoring position during the series.
"They just outplayed us," Price said. "Some of the things that happened in the series (were) unfortunate."
NOTES: Reds general manager Walt Jocketty said he would like to add a bat before the July 31 trade deadline. His preference would be someone who can play multiple positions. "Eventually we're going to get Joey and Brandon back, so you don't want to limit yourself," Jocketty said. ... Frazier, an All-Star third baseman this season, made his eighth start of the year at first and robbed Teixeira of a first-inning hit with a diving stop at the bag. ... Cincinnati is 3-8 in interleague games. ... Schumaker is 8 for 21 (.381) against Kuroda. ... RHP Mat Latos (2-1, 2.79 ERA) starts Monday night in Milwaukee against RHP Wily Peralta (10-6, 3.72). Latos left his most recent outing July 11 against Pittsburgh with lower back spasms. ... Cincinnati has won all three series against the Brewers this season. ... INF Jack Hannahan (right shoulder surgery) went hitless in his first 10 at-bats after his rehab assignment was transferred to Triple-A Louisville.