Cash was born in Justiceburg, Texas, and attended San Angelo Junior College, where he played football as well as baseball; he was drafted by the Chicago Bears, but declined to play pro football. After signing with the Chicago White Sox in 1955, he spent 1957 in the military and made his debut with the team in 1958, seeing limited play as an outfielder and pinch hitter. He appeared in 58 games for the 1959 pennant-winners, but the midseason acquisition of Ted Kluszewski left him on the bench, and he was hitless in four pinch-hitting appearances in the World Series. In December of that year he was traded to the Cleveland Indians in an eight-player deal that brought Minnie Miñoso back to Chicago, but the Indians dealt him to Detroit the following April in what turned out to be one of the worst transactions in baseball history. Not realizing what he had, Indians general manager Frank Lane traded Cash for Steve Demeter, who would play only four more games; both Chicago and Cleveland were haunted by Cash for the next 15 years, as he won a batting title and a World Series ring in a Detroit uniform.

Cash filled the middle of the Tigers lineup for 15 seasons as part of one of the sport's top offenses. He enjoyed his breakout season in 1961, leading the AL with a .361 average (Kaline finished second) and 193 hits, as well as collecting 41 home runs (6th in the AL), 132 runs batted in (4th), 119 runs scored (4th), 124 walks (2nd) for an .488 on base percentage (1st), and 354 total bases (2nd) for a .662 slugging average (2nd); but his season was overshadowed by the 61 home runs of Roger Maris, and teammate Rocky Colavito finished with more home runs and RBI. Still, his .361 average would be the highest by any major league player in the 1960s. The Tigers finished 101-61 for their best regular season record since 1934, and scored the most runs in baseball, though they finished second in the AL, eight games behind the New York Yankees; Cash was 4th in the MVP voting. Many observers have attributed some of the gaudy hitting statistics of 1961 to the league expansion that year, which resulted in a dilution of pitching talent; in addition, Cash later admitted to using an illegal corked bat during the season, and all of the above marks turned out to be career highs which he rarely approached again - in later years, he never reached 100 runs or 100 RBI, and never batted above .283. His 118-point drop to a .243 average in 1962 was the largest ever by a batting champion.

Even on his own team, Cash was overshadowed by his future Hall-of-Fame roommate Kaline. But although his batting average fell off sharply after 1961, Cash hit 30 or more homers four more times, and at least 20 in ten of the next eleven seasons; he also finished second in the league in home runs three times (1962, 1965, 1971), with the Tigers finishing among the AL's top three scoring teams every year from 1961 through 1968. He was considered one of the better defensive first baseman of the 1960s, leading the league in putouts (1961), fielding percentage (1964, 1967) and assists (1965-67). In addition, Cash hit .385 (10-26) with one home run in the Tigers' victorious effort in the 1968 World Series; with 2 out in the 7th inning of Game 7, Cash singled to start a 3-run rally that broke a scoreless tie and propelled the team to its first title since 1945. He later hit a home run to give Detroit a 1-0 lead in Game 1 of the 1972 American League Championship Series, though the Tigers went on to lose both the game and the series. He was released by the Tigers in August 1974 after hitting .228 in 53 games.

Cash was a career .271 hitter with 377 home runs, 1103 RBI, 1046 runs, 1820 hits, 241 doubles, 41 triples, 43 stolen bases, a .374 on base percentage, and a .488 slugging average in 2089 games. He holds Tigers career defensive records at first base in games (1912), putouts (14,926), assists (1303), and double plays (1328), having broken the marks set by Hank Greenberg and Rudy York.

Apart from his batting accomplishments, Cash was a favorite with his teammates, the media, and Tiger fans. He was known for his sense of humor; in one famous episode, while he was a baserunner, he attempted to call time out when he was about to be tagged out. Cash was also noted for never wearing a helmet during his entire playing career, being one of the few players who were permitted to do so after they were mandated in 1971. Also, in another famous incident long remembered by Tiger fans, on July 15, 1973 during a game at Tiger Stadium where California's Nolan Ryan was dominating on his way to his second career no-hit performance, as a joke Cash brought a table leg to the plate instead of a bat with two outs in the 9th inning (he had struck out 3 times already that night).

Cash was a broadcaster for ABC baseball broadcasts in 1976, and also was an announcer for Tigers cable broadcasts for several years. He drowned in a fatal accident off Beaver Island in northern Lake Michigan when he slipped while aboard a boat, fell, and struck his head. He was 51 years old.

On April 23, 2005, the high school and Little League baseball field in Post, Texas was dedicated in loving memory to the city's most famous athlete, Norman Dalton Cash.