Jones, 26, will receive $71.25 million over the life of the deal, with $47.5 million guaranteed, according to a source.
Talks between the two sides stalled last week but were "jump-started" Saturday morning.
"Julio is the type of young man we value, and this contract reflects that," said owner Arthur Blank.
First-year Falcons coach Dan Quinn added: "Julio represents all I love about football ... toughness and the way he goes about business and obviously very talented."
ESPN's Chris Mortensen contributed to this report.