Burrows was born in Sydney in 1937, a visiting flute virtuoso and teacher, Victor McMahon, played and inspired Burrows to take up the flute. By 1942 Burrows had taken up clarinet. He became well known in Sydney jazz circles and was playing in dance halls, nightclubs and radio bands.
During the 1960s and 1970s, Burrows was one of the best-known jazz musician in Australia and had many engagements in Australia and the United States.
In 1973 Burrows received the first gold record for an Australian jazz musician for his record Just the Beginning, started the first jazz studies program in the southern hemisphere, at the New South Wales Conservatorium of Music (under the direction of Rex Hobcroft) and was awarded the Member of the Order of the British Empire (MBE). In 1979 he was appointed Chair of Jazz Studies at the Sydney Conservatorium.
Burrows hosted the nationally televised show The Don Burrows Collection for six years. He has an extensive recording career in his own right with his groups and has performed on many more albums with other artists.
In the 1980s, Burrows was associated closely with the then young James Morrison.
Member of the Order of the British Empire (MBE), 1973
Officer of the Order of Australia (AO), 1987
Life member of the New South Wales Conservatorium of Music, 1988
Inducted into the ARIA Music Awards Hall of Fame, 1991
Named one of the Australian Living Treasures, 1989, 1999
Sir Bernard Heinze Award, for his service to Australia, 2000
Honorary Doctorate in Music, Sydney University, 2000
Honorary Doctorate in Music, Edith Cowan University, Perth, 2001
Ted Albert Award for Outstanding Services to Australian Music, Australasian Performing Right Association (APRA) Awards of 2004
Honorary Doctorate in Education, Central Queensland University, 2004
Queen Elizabeth II Silver Jubilee Medal (1977)
Inducted into the Australian jazz Bell Awards Hall of Fame, 2007