With scores locked at 2-2 at the bottom of the fifth inning, Ohtani delivered a two-run blast down right-center field from a Frankie Montas splitter in the zone.
The 398-foot home run was Ohtani's 29th of the season, boosting the two-way star's American League (AL) MVP case, with Judge likely to be his strongest opposition.
Judge produced his 434-foot solo shot over left-center field at the top of the eighth inning to halve the deficit and bring up his 50th home run for the season.
The slugger became just the seventh player to hit 50 home runs before September in AL or National League (NL) history.
Judge also became only the 10th player in MLB history to register multiple 50-home run seasons. Only three Yankees have achieved that feat; Judge, Babe Ruth (four) and Mickey Mantle (two).
"It's just another number," Judge said. "It's great, but I'm kind of upset about the loss."
The defeat leaves the Yankees with a 78-51 record having lost their last three games.
Pujols edges past Bonds all-time milestone
Pujols' two-run blast came in the third inning from Reds left-hander Ross Detwiler, bringing up his 694th-career home run.
The blast also tagged his major league record for the 450th different pitcher for a home run, breaking the all-time mark that had been tied with Bonds .
Padres overcome delays for key NL win
The San Diego Padres boosted their NL Wild Card aspirations with a hard-fought 6-5 win over the San Francisco Giants in a game full of delays, including an injury to home plate umpire Marvin Hudson as well as stadium lighting issues.
Brandon Drury's two-run first-inning home run set the Padres on their way before the delays, with three eighth-inning runs keeping the Giants alive, before Nick Martinez closed it out.