Pekka Rinne stopped 20 of 21 shots for the win. He had lost four of his previous five games against the Sharks.
The Predators rebounded from an overtime loss against Pittsburgh on Saturday.
Joe Pavelski scored for the Sharks, who have lost four of five since opening the season with four wins. Martin Jones had 22 saves.
Nystrom scored on a breakaway, squeezing the puck past Jones to his stick side. The goal, less than four minutes into the contest, snapped Nashville's two-game shutout streak in San Jose.
Pavelski scored his fourth goal on a rebound to tie it about 1 1/2 minutes into the third. He was knocked down twice in front of the net and got up just in time to slip the puck past Rinne.
Jankrok, who missed Saturday's game against the Penguins due to illness, went top shelf to beat Jones at 8:52 of the final period. It was his first goal of the season and his first point since the season opener.
NOTES: The Sharks recalled D Mirco Mueller and then scratched him. ... Predators D Seth Jones has three assists in his last three games. ... Sharks F Ben Smith (head) returned to action after missing three games. ... San Jose F Joonas Donskoi (lower body) was on the ice after missing five games. ... The Predators have yet to allow a first-period goal. ... Nashville plays at the Los Angeles Kings on Saturday. ... The Sharks travel to Dallas for a game against the Stars on Saturday.