Erne, who was a restricted free agent, signed a reported 1-year, $1.05 million deal with the Red Wings, according to CapFriendly. With Erne off the Lightning's books, the only player left for them to sign is superstar Brayden Point.
The 24-year-old appeared in a career-high 65 games for the Bolts last season, registering seven goals and 13 assists in a bottom-six role.
Tampa Bay now has nine picks in seven rounds for the 2020 draft.
The deal marks Steve Yzerman's first player trade as general manager of the Red Wings. Coincidentally, it came together with the team he managed from 2010-18.