Upon his graduation from Columbia Law, Justice Norcott worked as a Peace Corps volunteer in Nairobi, Kenya, where he was a lecturer in the faculty of law at the University of East Africa. He then served on the Bedford Stuyvesant Restoration Corporations Legal Staff in New York City and later as an Assistant Attorney General in the United States Virgin Islands. He was the co-founder and executive director of the Center for Advocacy, Research and Planning, Inc. in New Haven. Prior to his appointment to the bench, he also served as a hearing examiner for the Commission on Human Rights and Opportunities. He currently is an Associate Fellow of Calhoun College at Yale and serves as a lecturer there.

Justice Norcott was nominated to the Superior Court in 1979 and remained there until his appointment to the Appellate Court in 1987. In 1992, he was elevated to his current position as an Associate Justice of the Supreme Court. From July 2006 to August 2007, he served as Co-Chair of the Appellate Rules Committee.

His many civic involvements include membership on the University of New Haven Board of Governors (Emeritus); the Connecticut Bar Foundation; the Eastern Collegiate Football Officials Association; Omega Psi Phi Fraternity; and the Sigma Pi Phi, Beta Tau Boule.

Justice Norcott is the husband of Althea V. Norcott, and the father of Daryl, Tiffany and Candice Norcott.