He played in Major League Baseball (MLB) from 1955 to 1964 for the Cleveland Indians, Kansas City Athletics, and New York Yankees.

In June 1961, he was traded by Kansas City to the New York Yankees, becoming an impact pitcher as the Yanks won the 1961 World Series over the Cincinnati Reds.

Daley was purchased by the Cleveland Indians from the Sacramento Solons of the Pacific Coast League on August 18, 1955.

[2] He dropped his first major league start at Briggs Stadium in a 6–2 loss to the Detroit Tigers.

[6] On March 31, 1958, Daley was traded, along with Gene Woodling and Dick Williams, to the Baltimore Orioles, for Larry Doby and Don Ferrarese.

[9] Casey Stengel selected Daley as one of seven pitchers he picked for the American League All-Star team on July 2.

[15] Bob Cerv hit two home runs which assisted Daley in stopping a four-game winning streak by the Detroit Tigers, in May 1960.

[21][22] It is often rumoured that Daley was born naturally right-handed but developed polio as a child, "withering" his right arm.

He did suffer temporary paralysis to his right arm after birth due to forceps pinching a nerve.

Although he could barely raise it high enough to comb his hair,[2] Daley himself said that he has been living with it all his life without feeling handicapped a bit.

[22] In fact, Daley served as a fifth infielder, was competitive as a batter, shot golf in the low 80s, and was a proficient fisherman.