Draughn, a former undrafted free agent from North Carolina, also spent time with the Washington Redskins. He was released by the Chiefs during the final major roster cutdown this year.
Last season, Draughn rushed for 233 yards and two touchdowns. He also caught 24 passes for 158 yards.
"They told me I'm a special teams guy," Draughn said. "It's basically the same role I had last year in Kansas City, where I came in on third down, and just be ready when I'm called.
"It's a good thing. It's everything I've been praying for. It's a good team, happy to be a part of a winning tradition and good guys in the locker room from what I hear."