NEW YORK—Evan Longoria won the American League Rookie of the Year award Monday by unanimous vote, and Geovany Soto ran away with the NL honor.

Following a season of breakthroughs for the AL champion Rays, Longoria became the first Tampa Bay player to win a national award from the Baseball Writers’ Association of America. The power-hitting third baseman received all 28 first-place votes, becoming the AL’s first unanimous rookie winner since Nomar Garciaparra in 1997.

Soto, the steady catcher who helped the Cubs win the NL Central title, got 31 of 32 first-place votes. Th...

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