Nashville took the lead at PPG Paints Arena through Pontus Aberg, before Jake Guentzel levelled for the hosts.
After a goalless second period, Guentzel scored again before two goals inside 15 seconds - from Scott Wilson and Evgeni Malkin - sealed victory.
Guentzel has now scored 12 play-off goals in his rookie season.
The 22-year-old is two short of the record set by Dino Ciccarelli in 1981.
Game three of the series is in Nashville on Saturday (01:00 BST on Sunday).
Pittsburgh Penguins scored three goals in the final period to beat Nashville Predators 4-1 and take a 2-0 lead in the best-of-seven Stanley Cup final.