The seven-year NFL veteran has racked up 4,875 yards and 22 touchdowns rushing and hauled in 133 catches for 970 yards and a pair of scores over 84 games in his career with the Dallas Cowboys and the Seahawks.
Jones was released by Seattle this season after gaining 30 yards on 12 carries.
The Saints released 46-year-old kicker John Carney, the second all-time leading scorer in Saints history with 768 points. The move came two weeks after Carney was re-signed to replace Garrett Hartley, who had missed a potential game-winning field goal try against Atlanta.
Carney blew a short field goal attempt that would have given the Saints a late lead in Sunday's 30-20 loss to the Arizona Cardinals. He hit five-of-six field goal tries in his third stint with the team.