Brazelton was born in Waco, Texas. He graduated in 1940 from Princeton and in 1943 from the Columbia University College of Physicians and Surgeons in New York City, where he accepted a medical internship at Roosevelt Hospital. From 1945, after war service in the U.S. Navy, he completed his medical residency in Boston at Massachusetts General Hospital (MGH) before undertaking pediatric training at Children's Hospital.

He entered private practice in 1950, in Cambridge, Massachusetts. His interest in child development led to training in child psychiatry at MGH and the James Jackson Putnam Children's Center. He subsequently served as a Fellow with Professor Jerome Bruner at the Center for Cognitive Studies at Harvard University, then combined his interests in primary care pediatrics and child psychiatry and in 1972 established the Child Development Unit, a pediatric training and research center at Children's Hospital in Boston. Since 1988, he has been Clinical Professor of Pediatrics Emeritus at Harvard Medical School.

Dr. Brazelton was president of the Society for Research in Child Development (1987—1989), and of the National Center for Clinical Infant Programs (1988—1991). He has appeared many times before Congress committees in support of parental and medical leave bills, and continues to work with the Alliance for Better Child Care for a more comprehensive day care bill. He is a co-founder of Parent Action and serves on the National Commission on Children.

Dr. Brazelton has appeared several times on The Oprah Winfrey Show.

Dr. Brazelton's foremost achievement in pediatrics and child development has been to increase pediatricians' awareness of, and attention to, the effect of young children's behavior, activity states, and emotional expressions on the ways their parents react to, and thereby affect them. For example, one of his first publications in the field of psychology was a study with Kenneth Kaye of the interaction between babies' sucking at breast or bottle and the mother's attempts to maintain it, the earliest form of human "dialogue". The Brazelton Neonatal Behavioral Assessment Scale (NBAS) assesses not only the physical and neurological responses of newborns, but also their emotional well being and individual differences.

Brazelton has a large extended family in Texas. His daughter Kitty Brazelton is an eclectic music composer and professor in New York City.

On January 12, 2009, Brazelton passed on an urban legend about cats smothering babies in his newspaper column as fact and advised expectant mothers to get rid of their cats. This outraged many animal experts, who tried to dispel this myth.