Forsling scored at 1:02 of the first period for a 1-0 lead. Reinhart sent the puck from behind net to Barkov in the right circle and he passed it across to Forsling for his third goal of the season.
"We wanted to have a good start to this road trip and were ready to play the right way here," Forsling said.
The Panthers visit the Seattle Kraken on Tuesday, the second of a five straight away from home.
Chinakhov tied it 1-1 at 7:06 with a shot from the high slot for his third goal in two games.
Verhaeghe made it 2-1 at 7:28 when he tipped in Reinhart's shot for his 13th goal.
"It was a bit of a grinder," Panthers coach Paul Maurice said. "It was difficult, it seems, for both teams to finish some of the opportunities they had. Both goalies made some good saves.
"We were pretty darn consistent. I don't think we had a lull. We had a handful of extended offensive zone shifts. They had a couple, but it was pretty consistent."
The Blue Jackets failed to capitalize on two power plays early in the third.
"It's 2-1, chances were pretty even after the first couple periods," Sillinger said. "We find a way to get that [power play] or get a bounce and tie it 2-2 and put some pressure on them and it changes the dynamic of the hockey game."