The blast over the wall in right field by captain Jeter was the fifth solo homer slugged by New York as it battled back from a 5-3 deficit at Yankee Stadium.
Cleveland had won Thursday's opener 10-2 at the new park.
"Sometimes the first one is the hardest one to get," Jeter told reporters.
"I don't hit very many home runs. They're going to come every now and then but this was a big one because we got a win."
Johnny Damon, Mark Teixeira, Melky Cabrera and Robinson Cano also collected home runs for the Yankees while Mark DeRosa did likewise for Cleveland.
Reliever Brian Bruney (2-0) picked up the win for the Yankees who improved to 6-5. Jensen Lewis (1-1), who struck out his first two batters before yielding the homer to Jeter on a 3-1 pitch, took the loss in relief for the 3-8 Indians.
Mariano Rivera stranded two baserunners in the ninth to register his third save of the season.
Joba Chamberlain started for New York but struggled through 4 2/3 innings, giving up six hits, five walks and all five runs.
Phil Coke, Jonathan Albaladejo, Bruney and Rivera contributed 4 1/3 innings of shutout relief.
On Thursday, Yankees starter CC Sabathia left with the game tied 1-1 but New York relievers were clobbered for nine runs in the seventh inning.
"The bullpen came in and did a good job and that was the difference today," said Yankees manager Joe Girardi on Friday.
"You don't want to wait a long time. You want to win at home, you want it to be your place."
