One-dimensional multiferroic nanostructured composites have drawn increasing interest as they show tremendous potential for multifunctional devices and applications. Herein, we report the synthesis, structural and dielectric characterization of barium titanate ($BaTiO_3$)-bismuth ferrite ($BiFeO_3$) composite fibers that were obtained using a novel sol-gel based electrospinning technique. The microstructure of the fibers was investigated using scanning electron microscopy and transmission electron microscopy. The fibers had an average diameter of 120 nm and were composed of nanoparticles. X-ray diffraction (XRD) study of the composite fibers demonstrated that the fibers are composed of perovskite cubic $BaTiO_3$-$BiFeO_3$ crystallites. The magnetic hysteresis loops of the resultant fibers demonstrated that the fibers were ferromagnetic with magnetic coercivity of 1500 Oe and saturation magnetization of 1.55 emu/g at room temperature (300 K). Additionally, the dielectric response of the composite fibers was characterized as a function of frequency. Their dielectric permittivity was found to be 140 and their dielectric loss was low in the frequency range from 1000 Hz to $10^7$ Hz.