It's a sixth-round selection, according to NFL Network's Ian Rapoport, that can become a fifth-rounder depending on multiple conditions.
"I'm excited. I just got to process everything," Griffen told Josina Anderson. "I'm gonna reach out to (Lions running back) Adrian (Peterson) at some point."
The 32-year-old won't be available for the Lions' game against the Indianapolis Colts on Sunday due to the NFL's COVID-19 protocols. Detroit will have to pay half of Griffen's $6-million salary in 2020.
Griffen joined the Cowboys on a one-year free-agent deal in the offseason after a 10-year stint with the Minnesota Vikings.
The four-time Pro Bowler racked up 20 tackles and 2.5 sacks through seven games with Dallas. He thanked the club and head coach Mike McCarthy after learning of the trade, and showed excitement for his new team.
The Cowboys reportedly made teams aware of Griffen's trade availability after a disappointing 2-5 start this season.
Griffen will join a Lions pass-rushing unit that has struggled this season, ranking 28th in the league with just eight sacks. Defensive ends Romeo Okwara and Trey Flowers lead Detroit with four and two sacks, respectively.
The 3-3 Lions will face the Vikings in Griffen's expected Week 9 debut.