In this manuscript, we introduce a new $\rho$-th-order Kantorovich-type generalization of the modified Bernstein operators. First, we establish Voronovskaya-type theorems for the proposed operators. Then, we derive rates of convergence and establish global approximation theorems by means of the modulus of continuity, the $\mathbb{K}$-functional, the Ditzian-Totik weighted moduli of smoothness and the weighted $\mathbb{K}$-functional. Also, several numerical examples are presented to illustrate the approximation behavior using MATLAB. Finally, we construct a bivariate analog of these operators and investigate its convergence properties.