Forsling gave Florida a 1-0 lead at 1:02 of the first period. Reinhart sent a backhand pass from below the goal line to Barkov, who quickly got it over to Forsling in the left circle.
"We wanted to have a good start to this (five-game) road trip, and were ready to play the right way here," Forsling said.
Chinakhov tied it 1-1 at 7:06 with a shot from the high slot. It was his third goal in two games.
Verhaeghe responded for the Panthers to make it 2-1 at 7:28 when he tipped in Reinhart's shot.
"It was a bit of a grinder," Florida 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 had a chance to tie it on two power plays early in the third period, but they failed to capitalize.
"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."