Moore, the team's second-round pick in the 2009 NFL draft, had two years left on his deal. His salary-cap number would have been $6.525 million next season.
Durant, who signed a three-year deal with the Falcons before the 2015 season, had a cap number of $3,416,666 for 2016.
The release of Moore saves $3.35 million in cap space, while Durant's release equals a $2.58 million cap savings. By releasing Durant on Monday, the Falcons avoid having his $1.75 million base salary in 2016 becoming fully guaranteed.
Both Moore and Durant spent most of last season dealing with injuries. Moore, a one-time Pro Bowl selection, finished the season on injured reserve with an ankle injury after missing games earlier in the season with a groin injury. He played in 11 games this past season after playing in only seven the year before while dealing with multiple shoulder injuries and a foot injury.
"We want to thank both of these guys for their commitment and work ethic," coach Dan Quinn said. "They battled through injuries to give everything they had for their teammates this season, and I will always be appreciative of that."
Durant came to the Falcons with an injury history and missed three games this season due to various injuries. When healthy, he was arguably the team's best linebacker. He finished the season with 82 tackles, second on the team behind Paul Worrilow (95).
Moore admitted he had problems adjusting to the new style of defense implemented by Quinn in 2015. Kemal Ishmael, who replaced Moore at the end of last season, might get the first shot to earn the starting role going into next season.
As for Durant's spot at weakside linebacker, the Falcons are bound to address the void via free agency or the draft. The Falcons need a significant upgrade at linebacker, and Quinn knows what type of players he wants to target.
We certainly want to find ways to see our speed and tackling at linebacker [improve]," Quinn said. "I haven't dug in all the way with the group, but I have a real clear vision of what I'm looking for. It's the space tackling. In college football right now, there are so many spread offenses. So space and tackling, that's where so much of the game is.''