"It feels fine right now," Soriano said after getting struck in the ninth inning by a shot by Jason Kipnis that was headed right at his head. "It hit my glove or my hand, I'm not sure which one first."
Soriano recovered to get the ball and make the out, then the right-hander the next batter, Asdrubal Cabrera, on a soft liner for his 33rd save in 35 chances since replacing the injured Mariano Rivera as closer.
The Yankees have battled injuries all year and after being swept in three games in Chicago, felt the Tampa Bay Rays closing in on their AL East lead.
"We're really fortunate," manager Joe Girardi said.
New York took two of three from the Indians to open a four-game lead on the idle Rays.
"These are important games down the stretch," Girardi said. "We need wins and to win series. That's the way to do it, win series."
The Indians, meanwhile, keep losing. Kipnis had three hits and three stolen bases for Cleveland, which has lost nine of 10 and is 5-23 since July 26.
"We pitched OK, which made for a decent ballgame," Indians manager Manny Acta said. "That makes it easier to take than when you're trailing by a ton of runs and it sucks the energy out of everybody."
They did trail quickly, however, when New York took a 3-0 lead in the second off Ubaldo Jimenez (9-13). Granderson's 33rd homer in the sixth made it 4-2 and gave the Yankees a record eight current players with 200 or more career homers.
"It's cool to accomplish that," said Granderson, who had champagne in a bucket of ice awaiting him at his locker, where his nameplate was changed to read "200".
"When you see the list of names, all the guys together on this club, it speaks to what the organization is doing."
Boone Logan (5-2) pitched 1 2/3 innings for the win as Girardi turned to his bullpen early.
"They were well rested," Girardi said of Soriano, Logan and David Robertson, who worked 1 1/3 scoreless innings.
Nick Swisher had three hits as New York improved to 26-15 in series finales, including 10-5 in the deciding game of three-game sets.
Jeter went 0 for 5, but still leads the majors with 173 hits.
A large group of Yankees fans chanted "Der-ek Je-ter" each time New York's captain came to the plate. Indians fans responded with a resounding boo -- and each side turned up the volume throughout each at-bat.
Cleveland's fans cheered the loudest when Jeter bounced into double play to end the ninth against Esmil Rogers.
Garcia worked out of a bases-loaded jam in the third, getting Carlos Santana to pop a 3-2 pitch to right. Garcia thought he had Santana struck out on the previous pitch, stomping around the mound after plate umpire Gary Cederstrom called ball three.
The right-hander wasn't as fortunate in the fifth after getting two quick outs. Kipnis singled, Garcia hit Cabrera with a pitch and then walked Shin-Soo Choo. This time, Santana grounded a two-run single up the middle to make it 3-2.
Logan came on and retired Michael Brantley on a groundout with runners on first and third.
Later, Robertson stared in at Cederstrom and Girardi quickly met Cederstrom at the plate for a chat.
"I went out as peacemaker," Girardi said.
Granderson homered off reliever Tony Sipp. It was the 195th homer this year by the Yankees, who have gone deep in 40 of their last 45 games.
Garcia struck out six over 4 2/3 innings. He failed to go the necessary five innings for a starter to get the win and remained 5-1 in daytime starts this year.
Jimenez fanned four over five innings, yielding eight hits and three runs. The right-hander dropped to 1-6 with a 7.96 in nine starts since July 14.
Ohio native John Glenn, the former U.S. senator and astronaut celebrating 50 years since his historic space flight, tossed out the ceremonial first pitch. ... Girardi expects updates on 3B Alex Rodriguez, out since July 25 with a broken left hand, and LHP Andy Pettitte, sidelined since June 28 by a broken left ankle when the team returns home. Rodriguez could take regular batting practice before the Yankees play Toronto in New York on Monday. ... Granderson joined Rodriguez, Jeter, Swisher, Mark Teixeira, Raul Ibanez, Eric Chavez and Andruw Jones in the 200-homer club. ... Cleveland went 1-5 against the Yankees this season.