Data-driven discovery of governing equations has received growing attention in recent years. However, most existing methods are designed for scalar equations and suffer from the curse of dimensionality in high-dimensional systems, while failing to guarantee rotational and reflectional invariance. In this work, a Cartesian tensor–based sparse regression framework is proposed for the identification of governing equations expressed in tensor form. The core of the framework is the systematic construction of a complete and non-redundant Cartesian tensor candidate library, achieved through tensor combination, suffix assignment, and filtering. This construction inherently preserves invariance properties while effectively mitigating the combinatorial growth of candidate terms in high-dimensional settings. The proposed framework is validated on the two-dimensional Burgers equation, the two- and three-dimensional Navier–Stokes equations, and the three-dimensional Giesekus constitutive model. Across all test cases, the method consistently achieves lower identification errors and higher efficiency compared to the baseline method. These results demonstrate that the proposed approach provides a general and efficient framework for data-driven discovery of tensorial governing equations in high-dimensional systems.