Skowron was born in Chicago, Illinois, and was of Polish descent. His father was a city garbage collector. One day his grandfather gave the seven-year-old Skowron a haircut that resembled a known Italian dictator's, which resulted in his friends jokingly calling him "Mussolini", and caused his family to shorten the nickname to "Moose." The name stuck throughout his career.
Skowron attended Weber High School in Chicago, then went to Purdue University in Indiana, where he was a member of Tau Kappa Epsilon fraternity. Though Skowron went to the school on a football scholarship, he found himself better suited to baseball, hitting .500 as a sophomore in 1950, a record in the Big Ten Conference that lasted ten years.
Following his sophomore year at Purdue, Skowron was signed to play baseball for the Austin (MN) Packers in the Southern Minny League (Class AA-level town-team baseball). Skowron did so well in Austin that the Yankees made a contract offer.
He was signed by Yankees scout Lou Maguolo.
Skowron signed with the New York Yankees in September 1950 as an amateur free agent and played his first game for the Yankees on April 13, 1954. In the beginning, he was platooned at first base with Joe Collins, but from 1958 on he became the Yankees' full-time first baseman. He played in seven American League (AL) All-Star games as a Yankee: 1957, 1958, twice in 1959, twice in 1960, and 1961 (two All-Star Games were played in 1959 through 1962).
On November 26, 1962, he was traded by the Yankees to the Los Angeles Dodgers for Stan Williams. Although Skowron floundered against National League pitching, batting just .203 in 237 at bats with four home runs, he stunned his former team in the 1963 World Series, as he led the Dodgers with a .385 average and a home run, as Los Angeles swept New York in four straight games.
On December 6, 1963, he returned to the AL when he was purchased from the Dodgers by the then Washington Senators. On July 13, 1964, he was traded by the Senators to the Chicago White Sox. In 1965, he played in his eighth All-Star Game. On May 6, 1967, he was traded by the White Sox to the California Angels. He was released by the Angels on October 9, 1967.
He played in a total of 1,478 major-league games, all but 15 as a first baseman. (He was in 13 games as a third baseman and two as a second baseman.)
Skowron made the last out of the 1957 World Series, but the following year he knocked in the winning run in game six of the 1958 World Series. Skowron also hit a three-run home run in game seven to propel the Yankees to a World Series win, and a comeback from a 3-1 series deficit. He also scored the only run in game seven of the 1962 World Series against the San Francisco Giants.
During his time with the Yankees, he resided in Hillsdale, New Jersey.
In 1963, he appeared as himself in the Mister Ed episode "Leo Durocher Meets Mister Ed".
Skowron was once a playful target of his friend, Yankee pitcher Fritz Peterson. A known practical joker, Peterson was reportedly popular with his teammates, entertaining them with his elaborate jokes. He once used fake Baseball Hall of Fame letterhead to ask Moose Skowron to donate his pacemaker after he died
Skowron met and married Virginia Hulquist while he was playing for the Austin, MN Packers. He was inducted into the National Polish-American Hall of Fame on June 12, 1980 while living in Schaumburg, Illinois. In 1999, he became a community relations representative for the Chicago White Sox and was still holding this position when he died in 2012.
Skowron died at age 81 on April 27, 2012, in Arlington Heights, Illinois. He died of congestive heart failure after a long battle with lung cancer.