We consider the problem of tensor completion with graphs serving as side information to represent interrelationships among variables. Existing approaches suffer from several limitations: (1) they are often task-specific and lack generality or systematic formulation; (2) they typically treat graphs as static structures, ignoring their inherent dynamism in tensor-based settings; (3) they lack theoretical guarantees on statistical and computational complexity. To address these issues, we introduce a pioneering framework that systematically develops a novel model, theory, and algorithm for dynamic graph-regularized tensor completion. At the modeling level, we establish a rigorous mathematical representation of dynamic graphs and derive a new tensor-oriented graph smoothness regularization effectively capturing the similarity structure of the tensor. At the theory level, we establish the statistical consistency for our model under certain conditions, providing the first theoretical guarantees for tensor recovery in the presence of graph information. Moreover, we develop an efficient algorithm with guaranteed convergence. A series of experiments on both synthetic and real-world data demonstrate that our method achieves superior recovery accuracy, especially under highly sparse observations and strong dynamics.