With head coach Dan Quinn taking over the defensive playcalling duties this season, Sutton will assist him with in-game strategy, managing the clock, usage of timeouts and replay.
"Calling the defense, spending the extra time to get that right, and making sure our style and identity come across in the right way are exciting challenges," Quinn said in a statement. "Adding someone with Bob's knowledge and experience to assist with clock and game-management strategies will help us be the best version of the Falcons that we can be in 2019."
Sutton fills the role that was initially filled by Kyle Flood, who left to join Alabama's staff as offensive line coach.
The Chiefs fired the 68-year-old Sutton after they lost to the New England Patriots in overtime in the AFC Championship Game. He had been the team's defensive coordinator since 2013.
Before joining the Chiefs, Sutton spent 13 seasons with the New York Jets. He also was the head coach at Army from 1991 to 1999.