In the present paper, we define and study a new family of sampling-type operators. By composing C. Bardaro’s well-known generalized exponential sampling operators with Mellin differential and Mellin anti-differential operators of order m, we derive the m-th order Kantorovich-type exponential sampling series. This family of operators is highly general and encompasses, as special cases, the well-known exponential sampling Kantorovich operators. Here, the pointwise and uniform convergence of m-th order Kantorovich-type exponential sampling series is investigated. Additionally, quantitative estimates on the rate of approximation, asymptotic formulae, and Voronovskaya-type theorems are established. The derivation of these results relies significantly on certain algebraic moments of the associated kernels, which can be computed using the Mellin–Fourier transform (or, more concisely, the Mellin transform) and the well-known Mellin–Poisson summation formula. Thanks to the aforementioned results, the simultaneous approximation of a function and its Mellin derivatives can be addressed both qualitatively and quantitatively.