It's the second consecutive game the Dodgers hit a home run in the eighth inning off Miller (3-2), who prior to facing the Dodgers had not allowed a home run all year.
Miller was still in a state of disbelief following the Dodgers' 6-4 win over the Indians.
Puig's long-ball traveled 398 feet, according to ESPN's home run tracker.
Hernandez, batting for Chase Utley, hit a 1-2 pitch that barely cleared the wall in right field.
In Anaheim, New York pitcher CC Sabathia exited early with a strained left hamstring, and Eric Young Jr. rallied Los Angeles to an 11-inning victory that snapped the Yankees' six-game winning streak.
"You really have to shorten up your swing, and it was basically battle mode", he said.
"I left it in a spot where he can hit it pretty well".
In the bottom of the seventh inning, the Indians evened the score at 2-2 when third baseman Jose Ramirez hit a line-drive home run just inside of the foul pole in right field.
Later, in the top of the sixth, with the score 2-2 and the bases full of Dodgers, Andrew Miller walked to the mound from the bullpen to relieve Bauer and figuratively flipped Puig the middle finger by striking out the Los Angeles right-handed slugger as Puig swung futilely at Miller's nearly unhittable slider.
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"He's got nasty stuff", Bellinger said.
The Indians go into Thursday at 31-31 - the first time they have been.500 since April 18, when they were 7-7. I wasn't sure if it was going out.
Dodgers manager Dave Roberts said he didn't see the gesture and asked jokingly, "Are you sure it wasn't No. 1 he was signaling?" Get the app and get the game.
The Progressive Field crowd, perhaps sensing they were witnessing a preview of games with higher stakes four months into the future, broke into a loud cheer. Vargas gave up five hits with six strikeouts and one walk for his fourth straight win overall and second against the Giants this season.
Indians manager Terry Francona returned to the dugout after being hospitalized for a few hours Tuesday night.
Francona was taken by ambulance to Cleveland Clinic where he underwent several tests.
Dodgers outfielder Yasiel Piug earned a one-game suspension from Major League Baseball on Wednesday after a Tuesday incident where he made an obscene gesture toward fans in Cleveland.
Puig chose to appeal the suspension, and was in the lineup on Wednesday.
Puig flashed the middle finger on both hands at fans sitting behind home plate at Progressive Field after hitting a home run.