The Blackhawks outshot the Panthers 11-3 in the first period.
Brandon Montour gave the Panthers a 2-1 lead at 7:26 of the second period when he took a drop pass from Barkov in the high slot and shot it in off the post.
"There's a couple guys in front," Montour said. "I don't think [Fleury] was expecting it. I had a little bit of room. I took a quick wrister, and from what I saw, I don't think he saw it. It hit the post -- luckily for that one."
Weegar scored 50 seconds into the third period for a 3-1 Panthers lead. Carter Verhaeghe fed the puck to Weegar in the slot, and the shot deflected off the stick of Kirby Dach.
Caleb Jones brought Chicago to within 3-2 at 12:38 with a shot from the left circle on the power play. Kane got an assist on the goal, giving him 720 in the NHL and moving him past Denis Savard into second place in Blackhawks history.
Barkov scored an empty-net goal at 18:12 to make it 4-2, and Anton Lundell scored into an empty net at 18:51 for the 5-2 final.
"It was a good end to the road trip," Florida coach Andrew Brunette said. "It wasn't a pretty game, but we did enough to get a win. I think we've managed the game pretty well on the road of late. I thought at different times we got a little 'fire-wagony.' We have so much skill and can play at such a high level. Sometimes we get a little carried away. I think we did an excellent job this whole trip of playing road hockey, and whatever they give us, we're taking. We're not forcing issues. It's kind of grinding through it. In some of these games it wasn't maybe out best stretch of hockey, but we found ways to score and do the right things."