Chisenhall sent a double to center field in the second to score Carlos Santana, while Brantley's RBI single in the third drove home Jason Kipnis and Ramirez's RBI single in the fourth scored Abraham Almonte.
Alex Rodriguez homered in the fourth inning, but by then the Indians had already picked up their trio of runs off Yankees starter Ivan Nova. The Yankees put together a rally in the ninth against Tribe closer Cody Allen after Rodriguez led off with a single, stole second and then scored on a Carlos Beltran single. That made it 3-2.
"They were trying to mount something there," said Allen, who notched his 24th save. "They put some good swings on the ball. I hurt myself a little bit. I was getting behind in the count and I left some pitches over the plate. But, [thankfully], they hit some balls at some guys and we were able to get out of it and get a win."
Nova took the loss after a five-inning effort, while Tomlin picked up his first win of the season for the Indians in a one-run, two-hit performance. Tomlin threw 106 pitches through seven innings and struck out six.
Girardi said Nova never really could find a rhythm.
"He had to work really hard just to get through five innings and he was going through part of the order again -- some guys were giving him a little trouble," Girardi said. "I just felt to give us the best chance, I was going to go with [Adam] Warren [in the sixth]."
MOMENTS THAT MATTERED Escape act: After setting down the first six batters he faced in order, Tomlin surrendered a leadoff double to Headley in the third and later issued a one-out walk to Stephen Drew. The Yankees eventually moved runners to second and third, but the Indians starter escaped the early jam by inducing an inning-ending groundout off the bat of Brett Gardner. More >
"That was big," Tomlin said, "because you don't want to give those runs right back, especially early in the game."
Another one for A-Rod: In the fourth inning, Rodriguez sent his 26th homer of the season and 680th of his career over the left-field fence. The solo shot left A-Rod's bat at 107 mph and was projected to land 426 feet away, according to Statcast™. Rodriguez had been slumping, but he hit a grand slam Tuesday against the Twins before going deep again Thursday.
"I just think he's got it going again," Girardi said. "He's hitting the ball deeper, he's not out in front and he's driving the baseball."
Kipnis' strong night: Making his second start off the disabled list, following a bout with a right shoulder issue, Kipnis snapped out of an 0-for-17 funk with a three-hit night for the Tribe. In the third, the All-Star second baseman singled to center field and later came around to score on an RBI single from Brantley.
"He's a good hitter," Indians manager Terry Francona said. "The hope is a guy comes back and doesn't start off in one of those funks, but he was [good]. When he's hitting the ball the other way, that's a great sign for Kip."
"We're all pulling for him, myself included. He pitched his fanny off tonight. He threw a hanging breaking ball to A-Rod that he hit, but other than that, especially with all the lefties they had in the lineup, and in this ballpark, he pitched in enough and up. And then, [he] lived away and cut it and threw some curveballs. He just really pitched well." -- Francona, on Tomlin
WHAT'S NEXT Indians: For the second game of this four-game series on Friday in the Bronx, Indians right-hander Carlos Carrasco (11-9, 3.63 ERA) will take the ball in a 7:05 p.m. ET tilt at Yankee Stadium. Over his past four outings, Carrasco has spun a 1.36 ERA with 29 strikeouts, five walks and a .119 opponents' batting average over 33 innings, including a pair of nine-inning performances.
Yankees: The Yankees will give the ball to Masahiro Tanaka (9-5, 3.56 ERA), who is looking for his 10th win of the year. Tanaka is coming off a dominant performance in his last outing, a complete game in Toronto. He gave up just one earned run and five hits while striking out eight. Tanaka has only faced the Indians once in his career and took the loss last season.
Jordan Bastian is a reporter for MLB.com. Read his blog, Major League Bastian, follow him on Twitter @MLBastian and listen to his podcast. Grace Raynor is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.