Baltimore catcher Adley Rutschman scored the go-ahead run in the sixth inning on a wild pitch by Zack Britton, who left the game shortly thereafter with arm fatigue.
Kicking off their final home series of the regular season, the Yankees have already clinched the AL East division title.
Expectations were sky-high after Judge smashed his 61st home run of the year to match Roger Maris' 1961 single-season American League home run record on Wednesday in Toronto.
He approached the plate to deafening cheers from 47,583 fans in the first inning but struck out. The intensity of the moment was enough to rattle even the battle-tested manager Aaron Boone.
"It was a pretty electric atmosphere for his first at bat," said Boone. "I actually found myself kind of nervous for his first at bat, maybe just being all focused on it."
Fans' cheers turned to boos when Orioles reliever Felix Bautista intentionally walked Judge in his final at-bat in the eighth inning, after Yankees infielder Isiah Kiner-Falefa stole second.
"We're going to play to win the game," said Boone. "We all want to see Judge do it. I think they're trying to attack him within the context of the game."
While home run number 62 would fall far short of Barry Bonds' overall Major League Baseball (MLB) record of 73 homers hit for the National League's San Francisco Giants in 2001, Judge's race for the American League record has emerged as one of the most compelling storylines this season.
He has two more chances to break Maris' mark at home before the Yankees close out their regular season with a four-game roadtrip against the Texas Rangers.
