Blake Wheeler also had a third-period goal for the Jets (3-3).
Tim Brent scored in the second period for Carolina (1-3).
Kane tied it with a power-play goal at with 9:15 left, Wheeler gave the Jets the lead with 8:11 remaining, and Kane added an empty-net goal in the final minute.
"I thought we played a good game," Kane said. "It was a little bit slow getting on the board, but I thought we had enough chances. It could have been a little different."
"We wanted to get a couple of games under our belt before Oct. 9 when the real season starts and I think we made a nice step today," Wheeler said.
Brian Boucher made 35 saves for Carolina.
Carolina went 0 for 5 on the power play, while the Jets were 1 for 6.