There are numerous models in the form of ordinary differential equations that require numerical treatment, for they have no closed-form solutions due to some reasons, including non-linearity, stiffness, singularity, and stability. Having in mind the increasing demand for efficient and cost-effective algorithms yet straightforward to code; an attempt is made in this research paper to introduce a new time-efficient optimal exponentially fitted numerical algorithm whose qualitative analysis unfolds its local truncation error, zero-stability, A-stability, consistency, convergence, and accuracy (at least third-order). The proposed algorithm is also implemented in an adaptive step-size mode, whereas both constant and adaptive approaches outperform several existing algorithms when tested upon differential models taken from applied sciences. The efficiency curves further confirm the time-effectiveness of the proposed exponentially fitted one-step optimal method.