Most of the existing incomplete multi-view clustering (IMVC) methods focus on attaining a consensus representation from different views but ignore the important information hidden in the missing views and the latent intrinsic structures in each view. To tackle these issues, in this paper, a unified and novel framework, named tensorized incomplete multi-view clustering with intrinsic graph completion (TIMVC IGC) is proposed. Firstly, owing to the effectiveness of the low-rank representation in revealing the inherent structure of the data, we exploit it to infer the missing instances and construct the complete graph for each view. Afterwards, inspired by the structural consistency, a between-view consistency constraint is imposed to guarantee the similarity of the graphs from different views. More importantly, the TIMVC IGC simultaneously learns the low-rank structures of the different views and explores the correlations of the different graphs in a latent manifold sub-space using a low-rank tensor constraint, such that the intrinsic graphs of the different views can be obtained. Finally, a consensus representation for each sample is gained with a co-regularization term for final clustering. Experimental results on several real-world databases illustrates that the proposed method can outperform the other state-of-the-art related methods for incomplete multi-view clustering.