Judge, 25, became the first Yankee player since Derek Jeter in 1996 to pick up the American League award after a dazzling debut season.
Judge blasted 52 home runs during the regular season, breaking Mark McGwire's rookie record of 49 homers in the process.
His tally also surpassed Babe Ruth's Yankees record for home runs hit at home in a season.
As well as his home run exploits, Judge scored 128 runs and had 114 RBIs. The Yankees star also won the Home Run Derby in Miami in July.
In the National League, Bellinger was also a unanimous pick in the awards voted on by the Baseball Writers Association of America.
Bellinger, 22, smacked 39 home runs in his debut season for the Dodgers, who lost in game seven of the World Series to the Houston Astros earlier this month.
Bellinger set an unwanted record in the postseason with 28 strikeouts, including 17 in the World Series.
Paul DeJong of the St. Louis Cardinals was second in the voting with the Pittsburgh Pirates' Josh Bell third.