The deal is for $8 million and could increase to $10 million with incentives, a source told ESPN NFL Insider Adam Schefter.
Poe, 26, who has been selected twice for the Pro Bowl, totaled 10 1/2 sacks in 2013 and 2014 and has 13 in his five-year career.
"Poe is an athletic, big guy that can push the pocket," Falcons general manager Thomas Dimitroff said in a statement. "He will add to our defensive line group and will mix nicely with [Vic] Beasley, [Grady] Jarrett, Shelby and [Adrian] Clayborn. We are happy to be able to add him to our roster."
The Chiefs decided against using the franchise tag on Poe, despite having the option to do so after signing safety Eric Berry to a long-term contract. They would have had to offer Poe a one-year deal worth $13,387,000 had they franchised him.
Perhaps expecting Poe to sign elsewhere, Kansas City signed former Eagles defensive tackle Bennie Logan earlier this week.
Poe joined the Chiefs as their first-round draft pick in 2012. He was not as productive last season as he had been earlier in his career, when he was a consistent force against the run and strong as an inside pass-rusher.
He has been a workhorse for the Chiefs, missing only two games in his career because of injury. He also played a lot of snaps for a defensive lineman, but the Chiefs usually found him too valuable in both running and passing situations to provide him much relief.
Poe is athletic for a 346-pound player. The Chiefs put his athleticism to use occasionally in offensive goal-line situations. He twice scored on running plays in his career and threw a touchdown pass in a Week 16 victory last season against the Denver Broncos.
ESPN's Adam Teicher contributed to this report.