The agreement is worth $30 million over the five years, including $11 million guaranteed, a source told ESPN NFL Insider Adam Schefter.
The Falcons have been in a win-now mode since trading up in the draft to get receiver Julio Jones. They made it clear they wanted to add an explosive player on defense as well.
Edwards fits that profile. At 26 years old, he's expected to complement John Abraham as a pass-rusher in the short term and become his eventual replacement.
Edwards recorded 16.5 sacks over the past two seasons in Minnesota, despite sharing the spotlight with talented Vikings defensive end Jared Allen.
Edwards, who uses boxing as part of his training routine, got into the ring during the lockout and won his professional debut.
The Vikings selected the 6-foot-5, 270-pound Edwards in the fourth round of the 2006 draft.
Pat Yasinskas is the NFC South blogger for ESPN.com. Information from ESPN NFL Insider Adam Schefter and ESPN.com AFC North blogger James Walker was used in this report.