Brandon Saad, Scott Hartnell and Matt Calvert also scored for the Blue Jackets, who extended their winning streak over the Sharks to three games.
Sergei Bobrovsky, pulled early from Columbus' 3-2 loss to Winnipeg on Saturday, bounced back with 41 saves. The Blue Jackets have won three of five since opening the season with eight straight losses.
Joe Pavelski scored twice for the Sharks, who have lost six of eight after opening with four consecutive wins. Martin Jones was lifted after stopping 17 of 21 shots through the first two periods. Alex Stalock made two saves in the third.
The Sharks went 0 for 4 on power plays.
Saad, acquired from Chicago in a high-profile trade during the offseason, scored his fifth goal — and second in two games — midway through the first on a power play. Planting himself in front of the net, he was in position to push the rebound of a shot by Brandon Dubinsky off Jones' arm and into the net.
Pavelski, who has goals in five straight, tied the score late in the first on a redirect of Justin Braun's long shot.
Hartnell netted his third goal in four games to put the Blue Jackets back in the lead midway through the second. He dug it out behind the net and made a nice reverse move to get a clear shot.
Pavelski responded 3 minutes later with his second goal of the night. Joonas Donskoi stole a clearing pass and fed the puck to Pavelski, who beat Bobrovsky one-on-one.
Jenner's fourth goal in five games put Columbus ahead to stay. Hartnell beat Brent Burns to the puck behind the net and sent it out to Jenner, who powered it over Jones' right shoulder. Murray added another goal 29 seconds later.
Calvert added an empty-net goal in the third.
NOTES: F Joel Ward was named the Sharks Foundation "Sharks Player of the Month" for October. ... F Bryan Lerg was recalled from the AHL before the game, his fourth recall in 15 days. D Karl Stollery was also recalled and was a healthy scratch. ... Blue Jackets D David Savard played in his 200th NHL game. ... Dubinsky has points in five of his last six games. ... Donskoi got his first career assist. ... Columbus C Ryan Johansen had his first points in six games. ... The Blue Jackets travel to play the Los Angeles Kings on Thursday night and the Anaheim Ducks on Friday night. ... The Sharks host the Florida Panthers on Thursday night.