Forsberg, a native of Ostervala, Sweden, jumped a face-off win by Penguins center Sidney Crosby for a backhand from outside the crease that tied it 1-1 with 1:10 remaining in the third.
Stamkos won it with a snap shot from the right face-off circle. It was his fourth goal this season.
Juuse Saros made 16 saves for the Predators (6-9-4), who had lost five in a row (0-3-2).
Evgeni Malkin scored, and Arturs Silovs made 28 saves for the Penguins (9-5-4), who have lost three straight (0-1-2) and five of six (1-3-2).
The teams will play again in Stockholm on Sunday (9 a.m. ET; FDSNSO, SN-PIT, NHLN, SN) in the 48th NHL regular-season game played outside of North America and the 19th in Sweden, the most of any country.
Malkin put Pittsburgh ahead 1-0 on one of its two shots in the second period at 13:49, banking a wrist shot from behind the net off the stick of Michael McCarron and in off the back of Saros for his fourth goal this season.
Nashville was unable to score on two breakaways.
Erik Haula had a short-handed chance at 8:51 of the first period, but was denied by Silovs with a right pad save.
Justin Barron left the penalty box and went backhand to forehand twice before Silovs fell forward and extended his left pad for a save at 7:33 of the second.