Murphy was fired over "philosophical differences and productivity" issues, the head coach said.
Richard Rodgers, the team's assistant special teams coach, will step into Murphy's role.
"I told Coach Rodgers what I wanted," Rivera said. "I wanted things simplified. I wanted to come back to fundamental basics and that's the direction we're going to head right now."
The 2-7 Panthers gave up a 76-yard punt return for a touchdown in a 36-14 loss to Denver Sunday.
Carolina plays Tampa Bay this Sunday.