Tensor completion aims at estimating missing values from an incomplete observation, playing a fundamental role for many applications. This work proposes a novel low-rank tensor completion model, in which the inherent low-rank prior and external degradation accordant data-driven prior are simultaneously utilized. Specifically, the tensor nuclear norm (TNN) is adopted to characterize the overall low-dimensionality of the tensor data. Meanwhile, an implicit regularizer is formulated and its related subproblem is solved via a deep convolutional neural network (CNN) under the plug-and-play framework. This CNN, pretrained for the inpainting task on a mass of natural images, is expected to express the external data-driven prior and this plugged inpainter is consistent with the original degradation process. Then, an efficient alternating direction method of multipliers (ADMM) is designed to solve the proposed optimization model. Extensive experiments are conducted on different types of tensor imaging data with the comparison with state-of-the-art methods, illustrating the effectiveness and the remarkable generalization ability of our method.