Elvis Merzlikins made 22 saves for the Blue Jackets (10-16-2), who won their previous two games.
"It's all about putting the puck in the back of the net," Columbus forward Sean Kuraly said. "When you don't do that, it's going to make it tough. It's a good team. They finished their chances and there's some we would have liked to have tonight."
Brandon Montour gave Florida a 1-0 lead at 9:04 of the first period with a shot from the top of left face-off circle with Tkachuk screening Merzlikins.
Tkachuk made it 2-0 at 3:47 of the second period with a shot from close range after taking a feed from Reinhart.
Barkov extended the lead to 3-0 at 19:16 on a rebound.
"We had two solid periods," Blue Jackets coach Brad Larsen said. "The first was really good. The second they had a few looks with their power play and stuff, but there wasn't a lot. It's tough to come out of those two periods down 3-0, let's put it that way."