The Minkowski functional is a series of geometric quantities including the volume, the surface area, and the Euler characteristic. In this talk, we consider the Minkowski functional of the excursion set (sup-level set) of an isotropic smooth random field on arbitrary dimensional Euclidean space. Under the setting that the random field has weak non-Gaussianity, we provide the perturbation formula of the expected Minkowski functional. This result is a generalization of Matsubara (2003) who treated the 2- and 3-dimensional cases under weak skewness. The Minkowski functional is used in astronomy and cosmology as a test statistic for testing Gaussianity of the cosmic microwave background (CMB), and to characterize the large-scale structures of the universe. Besides, the expected Minkowski functional of the highest degree is the expected Euler-characteristic of the excursion set, which approximates the upper tail probability of the maximum of the random field. This methodology is used in multiple testing problems. We explain some applications of the perturbation formulas in these contexts. This talk is based on joint work with Takahiko Matsubara.