The Falcons released the 34-year-old Abraham, the team leader with 10 sacks in 2012, on March 1, the same day they also released running back Michael Turner and cornerback Dunta Robinson.
The team replaced Turner by signing Steven Jackson to a three-year, $12 million deal March 14. Now Umenyiora joins Jackson as Atlanta's second major free-agent addition.
The Falcons scheduled a news conference with Umenyiora for Thursday.
The 31-year-old Umenyiora gives the Falcons a slightly younger replacement at defensive end, but his production has declined in recent years. He had only six sacks for the Giants in 2012, when he started only four of 16 games. He had 55 tackles, five for losses, and one forced fumble.
He had a career-high 14 1/2 sacks in 2005, when he was a first-team All-Pro selection. He has reached double figures in sacks only one of the last four years - 11 1/2 in 2010.
Umenyiora, 6-foot 3 and 255 pounds, was a second-round pick from Troy State by New York in 2003. He has 75 sacks, 31 forced fumbles and 13 fumble recoveries in his career and helped the Giants win two Super Bowl championships.
He set an NFL record with 10 forced fumbles in 2010 and a Giants record with six sacks in a 2007 win over Philadelphia.
The presence of defensive ends Jason Pierre-Paul and Justin Tuck made Umenyiora a part-time starter last season, but he'll be expected to replace Abraham as the Falcons' top pass-rusher. Defensive end Kroy Biermann was second on the Falcons with only 4 sacks last season.
Umenyiora's agent, Tom Condon, couldn't be immediately reached for comment.
Umenyiora made a brief reference to his new NFL home on his Twitter feed when he tweeted "(hash)RISE UP" - the Falcons' slogan.