"In many ways, Clements helped define what a public servant is," the governor said. "He did his job quietly and intently."
In addition to his wife, Clements leaves two daughters, Rachel and Sara, the governor's office said.
Clements' family released a statement through that office, thanking everyone for their support and concern.
"Our family has lost a devoted husband and a beloved father," it read.
"There are no words at this time to describe our grief and loss. We thank our friends and those praying for us here and across the nation. Your well-wishes and prayers bring us strength. We appreciate your continued respect for our privacy during this terrible loss."
Hickenlooper appointed Clements as chief of the state's prison system in January 2011, according to Clements' online state biography. Before that post, he worked for 31 years in the Missouri Department of Corrections.
From October 2007 until January 2011, Clements was the director of adult institutions for the department, the biography says, overseeing 21 adult prisons.