Rosen was a .285 career hitter, with 192 home runs and 717 RBIs in 1,044 games. He was selected for the All-Star Game every year between 1952 and 1955. Rosen appeared on the cover of Sports Illustrated in 1955.
Rosen was one of the best all-time Jewish baseball players, along with Hank Greenberg (with whom he became close friends; Greenberg was in the front office with the Indians in the final years of Rosen's career) and Sandy Koufax.
Rosen was an asthmatic child, which had prompted the family's move south. His two favorite players were Lou Gehrig and Hank Greenberg. Rosen attended the University of Florida in the September session of 1941–42, where he played third base for the Gators.
Rosen enlisted after the bombing of Pearl Harbor, and spent 2 years in the Navy fighting in the Pacific during World War II, delaying his professional baseball career.

Rosen played for the Oklahoma City Indians in , and had one of the finest individual seasons in league history. He led all hitters in average (.349), hits (186), doubles (47), extra-base hits (83), RBIs (141), total bases (330), slugging percentage (.619), and on-base percentage (.437). He was elected Texas League MVP. 
In Rosen was Rookie of the Year for Triple-A Kansas City in the American Association.
In his first full season, in , Rosen led the American League in home runs with 37, hitting more than any previous American League rookie. He also averaged a league-best homer every 15.0 at bats, and led the league as well in HBP (10). He batted .287 and had 116 runs batted in, while coming in 5th in the league with 100 walks and a .543 slugging percentage. Despite his home run title, he only came in 17th in the American League MVP Award voting.
In he led the league in games played, and was 5th in the league in RBIs (102), extra-base hits (55), and walks (85). He batted .265, with 24 home runs.
Rosen led the American League with 105 RBIs and 297 total bases in . He also was 3rd in the league in runs (101) and slugging percentage (.524), 5th in hits (171) and doubles (32), 6th in home runs (28), and 7th in batting average (.302). Still, he only came in 10th in the American League MVP Award voting.
In , Rosen led the American League in home runs (43), runs batted in (145), runs (115), slugging percentage (.613), and total bases (367). He also came in second in OBP, and third in hits (201). He tied for 8th with Mickey Mantle in stolen bases. Defensively, he had the best range factor of all third basemen in the league (3.32), and led it in assists (338) and double plays (38).
He batted .336, and missed winning the batting title (and with it, the Triple Crown) on the last day of the season -- by only 1 percentage point, coming in second to Mickey Vernon. In Washington, the Senators heard that day that Rosen had made an out in his last at-bat, after getting 3 hits earlier in the game. Vernon was still ahead. The Senators conspired to make sure Vernon didn't have to bat again. Mickey Grasso doubled and let himself get picked off. Then Keith "Kite" Thomas singled, and was out when he ran leisurely to second.
He finally won the American League MVP Award in . It was by an unprecedented unanimous vote.
In , while batting .300, he led the league in sacrifice flies with 11, was 4th in SP (.506), and 5th in home runs (24), RBI (102), and obp (.404). He also hit consecutive home runs in the All-Star game despite a broken finger, and was named MVP of the All-Star Game.
Casey Stengel said of him: "That young feller, that feller's a ball player. He'll give you the works every time. Gets all the hits, gives you the hard tag in the field. That feller's a real competitor, you bet your sweet curse life."
Cleveland won the pennant, but lost to the New York Giants as Dusty Rhodes hit a home run off of Bob Lemon to win it, and give the Giants their last world title. It was his 5th straight year with 100 or more RBIs. Still, after the season Cleveland cut his $42,500 salary to $37,500.
In Rosen was in the top 10 in the league in at-bats per home run, walks, and sacrifice flies.
Back problems and leg injuries caught up with him, and Rosen retired at the age of 32 in 1956.

Rosen reentered baseball 20 years later as a MLB executive. He was president (and chief operating officer) of the Yankees (1978-79), then the Astros (1980-85), and then president and general manager of the Giants (1985-92). His maneuvering brought San Francisco from last place in to the NL West title in . 
Rosen's wife of 19 years, the former Teresa (Terry) Ann Blumberg, died on May 4, 1971, when she fell to her death from the Warwick Hotel in Philadelphia under mysterious circumstances. Rosen has 3 sons: Andy, Robert, and Jim. Andy Rosen is a musician who sings under the name of Goat; he resides in New York City with his wife Ana, who works for the Rainforest Alliance. Robert resides in Orlando, Florida, with his wife Linda and is the President of the Southeast Division of Deluca Homes; they have 2 children, Steven and Lauren Rosen. Jim Rosen resides in Los Angeles with Rosen's youngest grandchild, Zachary.
Rosen and his second wife, Rita (nee Kallman), have been married for over 30 years. Rosen occasionally consults for baseball teams, including a stint with the Yankees as special assistant to the general manager in 2001 and 2002.
In 1951, the television impresario Ed Sullivan, in his popular newspaper column, wrote about Rosen: "Of Jewish parentage, he is Catholic. At the plate, you'll notice he makes the sign of the cross with his bat." Enraged, Rosen insisted on a full and public retraction, pointing out that the mark he always made with his bat was the letter "x." 
Once a White Sox opponent called him a “Jew bastard.” Sox pitcher Saul Rogovin, also Jewish, remembered an angry Rosen striding belligerently to the dugout and challenging the “son of a bitch” to a fight. The player backed down. 
Rosen challenging an opposing player who had “slurred [his] religion” to fight him under the stands. When a Red Sox catcher called Rosen anti-Semitic names, he called time and “started toward him, to take him on.” Hank Greenberg recalled that Rosen “want[ed] to go into the stands and murder” fans who hurled anti-Semitic insults at him.
During his career, Rosen declined to play on the High Holy Days.
In an article in 1976 in Esquire magazine, sportswriter Harry Stein published an "All Time All-Star Argument Starter," consisting of five ethnic baseball teams. Rosen was the third baseman on Stein's Jewish team.
Member of the National Jewish Sports Hall of Fame (1980).
Member of the International Jewish Sports Hall of Fame.
Member of the Texas League Hall of Fame ().