Nyquist, Johnny Gaudreau and Patrik Laine each had a goal and an assist, and Kent Johnson and Nick Blankenburg each had two assists for the Blue Jackets (14-30-2), who ended a three-game losing streak. Joonas Korpisalo made 23 saves.
"Still weren't perfect by any chance tonight," Nyquist said. "But I thought we managed the puck a little bit better. Got some timely kills and the power play came through for us and got us going there in the second.
"Our battle level was pretty good today. We won a lot of 50-50 battles that we need to do. All the things that we've talked about, we still can do better, but a step in the right direction."
Nico Sturm had a goal and an assist, Timo Meier and Nick Bonino scored, and Kaapo Kahkonen made 24 saves for the Sharks (14-23-9), who have lost six of eight.
This was the first of eight straight on the road for San Jose, including four more before the NHL All-Star break: against the Boston Bruins, Detroit Red Wings, Carolina Hurricanes and Pittsburgh Penguins.
"We all know, no offense to that team, but if you look at the other teams we play on this road trip, if we play like that we're going to get killed," Sharks coach David Quinn said.