In this paper, we investigate the cosmic evolution of a spatially homogeneous and anisotropic Bianchi type I universe filled with new holographic dark energy (NHDE) and cold dark matter (CDM) within the framework of General Relativity by considering both the components of the universe to be interacting with each other. To obtain the exact solutions of Einstein’s field equations, we consider two expansion laws: an exponential expansion and a power-law volumetric expansion. The evolutions of some parameters of cosmological importance are studied for both the models corresponding to the exponential expansion and the power-law volumetric expansion. We observe that in both the models the anisotropy parameter decreases as time evolves and tends to zero at late times. The model corresponding to exponential expansion behaves like CDM model and the model corresponding to power-law volumetric expansion behaves like quintessence holographic dark energy model at late time. We also compare the equation of state (EoS) and energy density of our interacting NHDE model with that of quintessence scalar field and establish a correspondence between them. The quintessence potential is reconstructed which depicts the observed accelerated expansion of the universe.