Abreu led all rookies in home runs (36), RBIs (107), hits (176), doubles (35), runs scored (80), OBP (.383), slugging percentage (.581) and OPS (.964).
"I don't have any words to describe this moment," Abreu said through a translator on the MLB Network telecast. "I consider myself a good hitter, and I guess I've proven it."
Abreu also took to Twitter to express his gratitude.
Since the Rookie of the Year Award was first given out in 1947, only four rookies have batted at least .300, hit at least 30 home runs and driven in at least 100 runs, and all have earned the honor.
Abreu also becomes the first rookie in MLB history to finish in the top five in all three Triple Crown categories.
Abreu, 27, is the first White Sox player to win the award since Ozzie Guillen did it in 1985.
Abreu's power did wane during the final two months of the season as he participated in a 162-game season for the first time in his career. In his native Cuba, Abreu never had more than 312 at-bats in a season for his Cienfuegos club. He had 556 at-bats in 145 games for the White Sox this past season.
"It was difficult to get adjusted to a new country especially if you don't have your family with you," Abreu said Monday. "A lot of people really helped me make the adjustment coming to this country. Thanks to my wife, my mother, my father. The adjustment has been pretty good. I like it here."
Los Angeles Angels right-hander Matt Shoemaker and New York Yankees reliever Dellin Betances finished second and third, respectively, in the voting conducted by the Baseball Writers' Association of America, with Abreu earning all 30 first-place votes. Yankees ace Masahiro Tanaka finished fifth in the voting after his hot start was derailed by an elbow injury that held him out for most of the second half of the season. Houston Astros starter Collin McHugh finished fourth.
Information from ESPNChicago.com's Doug Padilla and The Associated Press contributed to this report.