After serving for four years in the U.S. Navy during World War II, Rosen played his entire ten-year career (1947–1956) with the Cleveland Indians in the American League (AL).

Following two decades as a stockbroker after retirement from baseball, Rosen returned to the game as a top front office executive in the late 1970s, serving the New York Yankees, Houston Astros and San Francisco Giants variously as president, CEO, and general manager.

[3] Rosen enlisted in 1942, and spent four years in the U.S. Navy fighting in the Pacific during World War II, delaying his professional baseball career.

Upon leading the Canadian–American League in home runs (16) and RBIs (86), while batting .323, however, he was bestowed his idol Hank Greenberg's nickname, "the Hebrew Hammer".

He was loaned to the Blues for the duration of the team's American Association season as part of a deal that sent reliever Charley Wensloff from the Yankees to the Indians.

In 1948, Rosen played most of the year in minor leagues with the Kansas City Blues, before joining the Indians in September.

[13] In the 2001 edition of the New Historical Baseball Abstract, Bill James named Rosen's 1953 season the greatest ever by a third baseman.

In 1954 he hit an even .300, led the league in sacrifice flies with 11, was fourth in SLG (.506), and fifth in home runs (24), RBI (102), and OBP (.404).

His five RBI in the game matched the record set by Ted Williams five years earlier, which still stood through the 2011 season.

In 1955 Rosen finished in the top ten in the league in at-bats per home run, walks, and sacrifice flies.

Tommy John thought he left because owner George Steinbrenner had replaced Bob Lemon, a friend of Rosen's, with Billy Martin as the team manager a month before.

[23] Rosen served as a supervisor of credit operations at Bally's in Atlantic City before a loan going sour led to him resigning.

[24] Rosen was hired two weeks after the 1980 season ended for the Astros by owner John McMullen, who in a controversial decision had fired Tal Smith despite forming a core that won the National League West title.

He was hired as president and general manager by Bob Lurie of the San Francisco Giants a week after he left the Astros, where he would serve until 1992.

Notably, one of his first actions was to remove the television and stereo systems from Candlestick Park as a means to have the players totally focused on baseball.

Through acquisitions (such as Rick Reuschel) and promoting draft picks such as Will Clark, Rosen and his maneuvering brought San Francisco from last place in 1985 to the NL West title in 1987 and the National League pennant in 1989, earning him the National League Executive of the Year honors.

Rosen occasionally consulted for baseball teams, including a stint with the Yankees as special assistant to the general manager in 2001 and 2002.

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.

[33] Hank Greenberg recalled that Rosen "want[ed] to go into the stands and murder" fans who hurled anti-Semitic insults at him.

A 2010 documentary, Jews and Baseball: An American Love Story, highlighted Rosen, who in it is frank about how he dealt with anti-Semitism: "There's a time that you let it be known that enough is enough. ...

Through 2014, he was fifth in career home runs (behind Sid Gordon), seventh in RBIs (behind Ryan Braun), and tenth in hits (behind Mike Lieberthal) among all-time Jewish major league baseball players.