Alex Nedeljkovic made 25 saves for the Sharks (38-34-8), who avoided being eliminated from Stanley Cup Playoff contention.
They moved three points behind the Los Angeles Kings, who played at the Seattle Kraken on Monday, for the second wild card from the Western Conference with two games remaining.
Both the Sharks and Predators would be eliminated with a Kings victory. Los Angeles has two games remaining after Monday.
San Jose won for the first time in 16 games against Nashville dating to Dec. 10, 2019.
Luke Evangelista scored twice, Ryan Ufko had two assists, and Justus Annunen made 20 saves for the Predators (38-33-10), who were 4-1-1 in their previous six. Nashville remained one point behind Los Angeles for the second wild card with one game remaining.
Chernyshov gave the Sharks a 1-0 lead on the power play at 9:53 of the second period. He scored on a one-timer from the left face-off circle off a pass from William Eklund.
Celebrini made it 2-0 at 3:04 of the third period, scoring with a one-timer from the slot off a pass from Chernyshov after Predators defenseman Justin Barron turned the puck over in the neutral zone.
Evangelista cut it to 2-1 on a breakaway at 15:39, beating Nedeljkovic on the glove side, but Celebrini added an empty-net goal to make it 3-1 at 18:15.
Evangelista’s second of the game made it a 3-2 final at 18:50. He scored in front on a deflection of Ufko’s shot from the right side with Annunen pulled for the extra attacker.
