The geophysical inversion problem is inherently underdetermined. Constrained inversion, by incorporating prior information, can effectively reduce the ambiguity in inversion results. A key research challenge lies in establishing a reasonable coupling mechanism between the prior information and the inversion model. Traditional cross-gradient coupling methods often exhibit weak coupling effects in regions with small model gradients. In this paper, we introduce a new coupling method called “minimum cross-gradient support” (MCGS), which enhances the balance between model gradient magnitude and the influence of gradient direction by applying a minimum support function to the cross-gradient. We evaluated the coupling effects of MCGS in comparison with two others coupling methods: normalized cross-gradient (NCG) and joint minimum gradient support (JMGS), the latter of which also employs the minimum support function. Theoretical models demonstrate that MCGS retains the advantage of reduced dependence on strictly accurate prior information, enhances constraint effectiveness in weak gradient regions, and the strength of constraints can be flexibly adjusted through focusing factors. To verify the effectiveness of MCGS, we conducted two synthetic experiments: a double-blocks model and a nappe structure model, applied to magnetotelluric (MT) constrained inversion. The results showed that MCGS constrained inversion achieved better performance than both non-constrained inversion and cross-gradient constrained inversion, with lower residual cross-gradient values and higher model recovery accuracy compared to the true model. Finally, we applied the proposed MCGS coupling to real data sets from the Junggar Basin. The inversion results revealed the resistivity structure of the sedimentary layer and imaged possible residual Carboniferous sediments beneath the Permian, as well as the distribution of the Paleozoic basement. These results provide valuable evidence supporting the subduction tectonic evolution model of the region.