Zach Werenski and Sean Monahan scored for the Blue Jackets (18-17-7), who had won four of six (4-1-1). Jet Greaves, who started his seventh straight game, made 31 saves.
It was Monahan’s first game after missing the previous four with an undisclosed injury.
"We missed a couple opportunities,” Columbus coach Dean Evason said. “We're giving up breakaways. It's not a good remedy for a win.”
Regenda gave the Sharks a 1-0 lead at 19:11 of the first period. Wennberg fed a 2-on-1 pass to a streaking Regenda, who finished over the glove of Greaves.
Wennberg’s power-play goal extended it to 2-0 at 4:14 of the second period. Wennberg received a pass in the left corner, went to the front of the net and beat Greaves five-hole on the backhand.
It was Wennberg’s first game since signing a three-year, $18 million contract ($6 million average annual value) with San Jose on Sunday.
"Obviously, for the confidence and just the fact to do that, but I mean, at the end of the day, I mean, it's all about the team," Wennberg said. "I'm just happy to find a way to win this game."
Werenski cut it to 2-1 at 19:15, gathering the puck in the right circle and scoring on a wrist shot over Nedeljkovic’s blocker.
"We didn't really have much in the first half of the game. We got some jump at the end of the second," Werenski said. "I didn't think our third period was bad."