Laine skated end-to-end and got around Chicago defensemen Calvin de Haan and Riley Stillman before tucking the puck past Lankinen at 11:32 to tie it 3-3.
"I saw of bunch of openings. Sometimes I just need to take it to the net more," Laine said. "The goal gave me confidence, and good things happen when you go to the net."
Laine scored three times in the two games after going 14 games without a goal.
"I felt like myself out there tonight," he said. "We took advantage of some big plays. I can make things happen on my own, we just need to string something together and get some wins."
Stillman, who was acquired in the same trade as Connolly, made his Blackhawks debut and was minus-1 with four hits and two blocked shots in 21:23.
"I thought Stillman was very physical, he was active, he got stops for us, was jumping in the play," Chicago coach Jeremy Colliton said. "I asked him to pull back a little bit, but he did good stuff out there."
NOTES: The Blue Jackets were playing their first game without captain Nick Foligno, who was traded to the Toronto Maple Leafs on Sunday. … Blackhawks defenseman Adam Boqvist left the game at 4:46 of the first period after being struck in the face by the puck off the stick of Blue Jackets forward Zac Dalpe. Colliton said Boqvist had concussion symptoms. … Kane scored 16 points (four goals, 12 assists) in the eight games against Columbus this season.