Edward Hammond Boatner, Jr. was born in Boston, Massachusetts, and grew up in Saginaw, Michigan. He had a musical background: his father, Edward Boatner, was a baritone singer, composer and college music professor; his brother was a classically trained pianist; and his mother was a piano teacher.
Sonny was given up for adoption in 1924 by his father. No one seems to know why Boatner gave his son away, but the child was adopted by the Stitt family, who raised him in Saginaw. He later began calling himself "Sonny". While in high school in Saginaw, Stitt played in the Len Francke Band, a local popular swing band.
In 1943, Stitt first met Charlie Parker, and as he often later recalled, the two men found that their styles had an extraordinary similarity that was partly coincidental and not merely due to Stitt's emulation. Parker is alleged to have remarked, "Well, I'll be damned, you sound just like me", to which Stitt responded: "Well, I can't help the way I sound. It's the only way I know how to play." Kenny Clarke remarked of Stitt's approach: "Even if there had not been a Bird, there would have been a Sonny Stitt".
Stitt had played in some swing bands in the early 1940s and was featured in Tiny Bradshaw's big band in the early forties. He replaced Charlie Parker in Dizzy Gillespie's bop big band in 1945 and in 1946 made the first recordings under his own name for Savoy Records, which established his bop credentials.
Stitt played alto saxophone in Billy Eckstine's big band alongside future bop pioneers Dexter Gordon and Gene Ammons beginning in 1945 when he started to play tenor saxophone more frequently, in order to avoid being referred to as a Charlie Parker imitator. Later on, he played with Gene Ammons and Bud Powell. Stitt spent time at the Federal prison in Lexington, Kentucky, between 1948 and 1949 for selling narcotics.