Traditional approaches to solving differential games, such as dynamic programming and the Pontryagin minimax principle, often rely on simplifying assumptions like infinite-horizon settings, which reduce the Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) to a time-invariant form. However, extending these methods to finite-horizon scenarios remains challenging due to the inherent time-dependent complexity of the HJI PDE and the lack of analytical control policies. This paper introduces a novel iterative algorithm grounded in the Pontryagin minimax principle to address finite-horizon differential games. The proposed method ensures convergence to valid solutions while adhering to input constraints.