Multispectral images (MSI) can help deliver more faithful representation for real scenes than the traditional image system, and enhance the performance of many computer vision tasks. In real cases, however, an MSI is always corrupted by various noises. In this paper, we propose a new tensor-based denoising approach by fully considering two intrinsic characteristics underlying an MSI, i.e., the global correlation along spectrum (GCS) and nonlocal self-similarity across space (NSS). In specific, we construct a new tensor sparsity measure, called intrinsic tensor sparsity (ITS) measure, which encodes both sparsity insights delivered by the most typical Tucker and CANDECOMP/PARAFAC (CP) low-rank decomposition for a general tensor. Then we build a new MSI denoising model by applying the proposed ITS measure on tensors formed by non-local similar patches within the MSI. The intrinsic GCS and NSS knowledge can then be efficiently explored under the regularization of this tensor sparsity measure to finely rectify the recovery of a MSI from its corruption. A series of experiments on simulated and real MSI denoising problems show that our method outperforms all state-of-the-arts under comprehensive quantitative performance measures.