The deal is for one-year and worth $6 million, sources told NFL Network's Ian Rapoport.
Griffen is a four-time Pro Bowler and notched eight sacks in 15 games last year. He spent the last 10 seasons in Minnesota.
The Seattle Seahawks and a reunion with the Vikings had been the most likely destinations for the 32-year-old prior to Wednesday, Rapoport added, but the veteran defender will contribute to a deep Cowboys front four.
Cowboys owner Jerry Jones also added Dontari Poe and Gerald McCoy to the interior of head coach Mike McCarthy's defensive line in 2020, while former second-round pick Randy Gregory could enter the mix if reinstated.
Griffen accumulated at least eight sacks in five of his last six campaigns and is also a stout run defender.