Chudzinski, 44, has spent three of his nine years as an NFL assistant coach with the Browns, including one season as tight ends coach and two years as offensive coordinator.
He also has had two stints on the staff of the San Diego Chargers and has been Carolina's offensive coordinator each of the last two seasons.
Since the Browns rejoined the NFL in 1999, their best offensive season came under Chudzinski in 2007. The team scored 402 points that year while going 10-6. The Panthers set club records for total yards (6,237) and first downs (345) during Chudzinski's first year as their offensive coordinator.
The new Cleveland coach played tight end at the University of Miami from 1986-90.
Chudzinski replaces Pat Shurmur, one of eight NFL coaches to be fired since the close of the regular season.
Kansas City has hired Andy Reid to fill its vacancy and Buffalo has selected Doug Marrone.
Arizona, Chicago, Jacksonville, Philadelphia and San Diego are still looking for a new coach. Jacksonville became the latest to create a vacancy when the Jaguars fired Mike Mularkey early Thursday.