Abreu was playing for Cienfuegos in Cuba, and deGrom was in A-ball for the second straight year after coming back from Tommy John surgery.
On Monday, the pair were runaway winners in balloting for rookies of the year.
The Chicago White Sox first baseman was voted the AL honour unanimously by the Baseball Writers' Association of America, and the New York Mets pitcher won the NL award in a landslide.
One year after defecting from Cuba, the 27-year-old Abreu had a .581 slugging percentage to become the first qualifying rookie to lead the major leagues since Oakland's Mark McGwire in 1987, according to STATS. Abreu was sixth in batting at .317, tied for fourth in home runs with 36 and fifth in RBIs with 107.
He said playing in the major leagues "never crossed my mind when I was a kid in Cuba."
"But in 2013, after the World Classic, I realized that I belonged to the major leagues and I am very happy that I'm here," Abreu said through a translator during a news conference in Chicago. "And I'm very happy that my family's happy that I am in the major leagues. Of course, now I realize that I could make it."
Abreu received all 30 first place votes for 150 points in balloting by the Baseball Writers' Association of America.
Los Angeles Angels right-hander Matt Shoemaker was second with 40 points, followed by New York Yankees reliever Dellin Betances (27), Houston pitcher Collin McHugh (21) and Yankees pitcher Masahiro Tanaka (16), who was hurt for most of the season's second half.
Abreu hit .383 with three homers and nine RBIs for Cuba at last year's World Baseball Classic, then defected that summer and signed a $68 million, six-year contract with the White Sox in October. He became the sixth White Sox player to earn the honour following Luis Aparicio (1956), Gary Peters (1963), Tommie Agee (1966), Ron Kittle (1983) and Ozzie Guillen (1985).
"Playing baseball in Cuba made me the whole player I am now," Abreu said. "I learned baseball in Cuba, and I'm very grateful of that."
DeGrom received 26 of 30 first-place votes and 142 points. Speedy Cincinnati outfielder Billy Hamilton was second with four firsts and 92 points. He hit .285 with 38 stolen bases in the first half, then slumped to .200 with 18 steals after the All-Star break.
"I was just thankful to be in the big leagues this year," DeGrom said.
A shortstop in college, deGrom's career went on hiatus for a year when he needed reconstructive elbow surgery in October 2010.
"I just tried to stick with the rehab program that I was given and follow it step by step," he said. 'Thankfully I had a pretty smooth rehab process and never had any setbacks, and I was on schedule the whole way."
He made it to Triple-A this year and was a month shy of his 26th birthday when he was called up to pitch out of the bullpen last spring. But an injury to Dillon Gee led to deGrom's debut as a starter against the Yankees on May 15.
"He was very much on our radar," Mets pitching coach Dan Warthen said. "We were going to find a spot for Jacob deGrom."
With his shoulder-length hair flapping out from under his cap, deGrom was a hit right from the start: His third-inning single in the 1-0 defeat ended an 0-for-64 start to the season at the plate for New York's pitchers, the worst slide to open a season in major league history.
DeGrom went 0-4 with a 4.39 ERA in his first seven starts, then won at Miami with seven scoreless innings on June 21. He went on to win nine of his last 11 decisions, compiling a 1.99 ERA, and finished 9-6 with a 2.69 ERA.
On Aug. 2, deGrom took a no-hit bid into the seventh inning against San Francisco before Pablo Sandoval's two-out double. He also left his mark against the Marlins on Sept. 15, striking out Miami's first eight batters to tie the record for whiffs at the start of a game. DeGrom joined Houston's Jim Deshaies in 1986 as the only pitchers to accomplish the feat since 1900.
DeGrom became the fifth Mets winner of the award, joining Tom Seaver (1967), Jon Matlack (1972), Darryl Strawberry (1983) and Dwight Gooden (1984).
AP Sports Writer Andrew Seligman in Chicago contributed to this report.