This paper introduces a novel computer-aided tuning (CAT) method for coupled-resonator microwave bandpass filters. The method is based on the estimation of the Jacobian of the function that relates the physical filter design parameters to the extracted coupling parameters. Lately commercial full-wave electromagnetic (EM) simulators provide the adjoint sensitivities of the S-parameters with respect to the geometrical parameters. This information leads to an efficient estimation of the Jacobian since it no longer requires finite difference based evaluation. The tuning method first extracts the physically implemented coupling matrix and estimates the corresponding Jacobian. Next it compares the extracted coupling matrix to the target coupling matrix (golden goal). Using the difference between the coupling matrices and the pseudo-inverse of the estimated Jacobian, a correction that brings the design parameters closer to the golden goal is obtained. This process is repeated iteratively until the correction becomes sufficiently small with respect to a user specified goal. In the case of coupling structures with multiple solutions, the Jacobian is calculated for each admissible solution. This paper presents a criterion to identify the physical solution among the different solutions. The CAT method is applied to the design of a cascaded triplet (CT) filter implemented in a microstrip technology. This filter is a well-known examples of a non-canonical coupling structure.