Ron Rivera's men went ahead with just under five minutes left in the first quarter, when Cam Newton ended a massive drive by somehow evading the rush and finding Greg Olsen, who wide open in the endzone, for a one-yard touchdown.
A 47-yard field goal from Rian Lindell narrowed the deficit by the end of the first quarter, but DeAngelo Williams took advantage of a gaping hole in the Bucs' defence to make a 12-yard rush and extend the visitors' lead to 14-3.
Lindell made it an eight-point ball game with another kick - this time from 48 yards - with four minutes remaining before half time.
It was Newton himself who extended the lead in the third quarter, rushing from six yards. He then threw to Mike Tolbert early in the fourth as the Panthers ran riot.