The 24-year-old Lindback played in 16 games (10 starts) this season for Nashville, going 5-8 with a 2.42 goals-against average and a .912 save percentage. The 6-foot-6 Lindback served as Pekka Rinne's backup.
"We are very pleased to announce this trade today," Tampa Bay vice president and general manager Steve Yzerman said. "We believe Anders will join the team and give us a nice lift at the goaltending position as we prepare for the 2012-13 season. He has the potential to develop into the No. 1 goaltender we covet."
Dwayne Roloson, who made 31 starts for the Lightning this season, is an unrestricted free agent.
Nashville received goalie Sebastien Caron, two second-round selections in this month's draft and a 2013 third-round pick from Tampa Bay.