Glaser was born in Spalding, Nebraska. He was the brother of fellow country musicians Chuck and Tompall Glaser. With his brothers, he was a member of the group Tompall & the Glaser Brothers. He was also a solo artist. In 1984, his single, "You're Gettin' to Me Again" reached No. 1 on Billboard's Hot Country Songs. He was named as Best New Male Vocalist by the Academy of Country Music in 1984.