Nabokov needed to make just one save in the first period, eight in the second and seven in the third for his second shutout of the season. He was involved in a 0-0 tie against the Minnesota Wild on October 6.
Jillson, a 21-year-old defenseman, gave San Jose a 2-0 lead with 2:41 left in the second period while on the power play. He one-timed Gary Suter's pass from the right point past goaltender Ron Tugnutt.
Marcus Ragnarsson also scored on the power play as the Sharks returned home from a disappointing road trip and avoided a three-game losing streak.
Owen Nolan collected two assists to tie Jeff Friesen's record for most career points in a Sharks uniform with 350 -- 163 goals and 187 assists.
Tugnutt finished with 19 saves for the Blue Jackets, who lost their fourth straight game and fell to 1-6-4 this season.