New York sent 10 men to the plate in a five-run first inning, providing Rogers -- who was claimed off waivers from the Blue Jays on July 31 -- with all of the support he'd need in his first Yankees start.
"Unbelievable; an unbelievable opportunity," Rogers said. "When you come in the first inning and they got five right away, I just told myself try to throw five or six innings and win the game."
Rogers worked five solid frames of one-run ball before Carlos Beltran blew the game open in the sixth with his 11th career grand slam and first as a Yankee, crushing a John Axford pitch into the lower deck in right field.
The Indians had intentionally walked Jacoby Ellsbury to load the bases for Beltran, who was 1-for-8 lifetime against Axford before that matchup.
"I faced John Axford many times in the National League; I guess I had maybe like one hit against him," Beltran said. "He felt that it was the right matchup for me. I was able to put a good at-bat and come through for the team."
Cleveland clawed back against the Bombers' bullpen, with David Huff allowing a run and Shawn Kelley charged with four, but Adam Warren, Rich Hill and Dellin Betances got the situation under control in recording the final six outs.
The victory moved the Yankees a season-high seven games over .500 at 61-54, having won 14 of 21 games coming out of the All-Star break. Yet there was an off-note, as Brian McCann sustained a concussion on a foul tip and left the game for a pinch-hitter in the sixth.
"I'm not sure what it means right now," manager Joe Girardi said. "We've got to go talk about it, obviously."
Beltran, Stephen Drew and Martin Prado knocked run-scoring singles in the big first inning against starter Trevor Bauer, which also included a Chase Headley bases-loaded walk. Bauer lasted just 3 1/3 innings, permitting six hits, four walks and a hit batsman.
"I threw some pitches that I thought were strikes that weren't, and I threw some pitches that clearly were nowhere close," Bauer said. "I ended up walking some people. That's a tough way to start the game off."
Derek Jeter logged the first of those hits on a grounder to shortstop Jose Ramirez that clanged off of first baseman Carlos Santana's glove, scored as a single. The hit was the 3,430th of Jeter's career, tying Honus Wagner for sixth place on the all-time list.
Ichiro Suzuki added a seventh-inning single that marked his 2,810th career hit, tying George Sisler for 48th place all-time.
Rogers appeared in 44 games for Cleveland in 2012 before being packaged to Toronto for Mike Aviles and Yan Gomes. He held his former club to a run on four hits in an 88-pitch outing. Santana logged Cleveland's lone RBI off Rogers with a first-inning single.
"This is what we've seen from him," Girardi said. "I just wasn't sure how much he would be able to give us, and I think we were pretty conscious of watching his stuff continue to be sharp as his pitch count mounted. He did a great job."
In his brief time with the team, Rogers said that pitching coach Larry Rothschild has encouraged him to lengthen his stride, using his back more to improve his fastball control.
"It's great. I think having him on the ballclub, he has a great arm and he's a very young guy," Beltran said. "Facing him when he was with Toronto, he's a guy that has a lot of movement on the ball. He just went out and gave us an opportunity to win the ballgame."
David Murphy lined a run-scoring single off Huff in the sixth to trim Cleveland's deficit. Ramirez greeted Huff with a liner to right field, but Ichiro saved at least one run with a leaping grab.
Kelley struggled mightily in the seventh, facing four batters and failing to retire any of them before Santana stroked a two-run double off Warren and Murphy lifted a sacrifice fly. In the ninth, Betances retired Nick Swisher with two men on, securing the Yanks' fifth win of the homestand.
"It's nice, but it got interesting again," Girardi said. "They had a chance to make it a one-run lead in the ninth if Swish hits a home run, but it was nice to get the runs. It was great for our pitchers because they're not always going to be perfect. Hopefully, we continue to swing the bats."