The fluid-dispensing system is used in a personal care appliance, such as a power toothbrush. The dispensing system includes a strip of discrete, fluid-containing packets, a compressible sealing member and a hollow needle for puncturing the packets, the hollow needle being connected to a workpiece such as a brushhead via a connecting line. An actuation assembly includes a piston which moves successive packets against the sealing member, initially sealing the packet thereto. Further pressure results in compression of the sealing member and puncturing of the packet by the needle, with fluid being forced by piston pressure from the packet into the hollow needle and then to the connecting line for delivery to the workpiece.