The Brown v. Board of Education of Topeka decision was announced on May 17, 1954. In my preliminary research it appears that the 1966-1967 Alcoa Tornadoes, Coached by the late Vernon Osborne "Coach O", were the first integrated team to win a state championship in basketball. This was back in the days of a single tournament for all teams.
During the 1965-1966 season the all black Pearl High School team out of Nashville, won the state championship.
In all previous years, the teams were all white.
1967 Alcoa integrated 1966 Pearl all black
1965 Murfreesboro all white
1964 Donelson all white
1963 Oak Ridge all white
1962 Bradley Central all white
1961 Oak Ridge all white
1960 Hampton all white
1959 Alcoa all white
1958 Lenoir City all white
1957 Linden all white
1956 Linden all white
1955 Linden all white
http://tssaasports.com/history/results/?sport=basketball-boys