This article presents an asymptotic prescribed output-feedback control method for robotic systems without function approximation techniques such as neural networks (NNs) and/or fuzzy logic systems (FLSs). The derived controller can not only drive the tracking error to zero within the predefined transient and steady-state boundary but also avoid the conventional recursive controller design procedure. To realize such objectives, a novel high-gain observer is first constructed by employing a sliding mode-like modification term, such that the finite-time (FT) convergence of the observer error can be achieved. Then, an adaptive proportional–integral approximation-free output-feedback controller is designed. Unlike known results, a tracking error surface is involved into the suggested control framework, which is dedicated to allowing one-step controller design procedure. This control is more straightforward to implement with reduced computational burden. Rigorous theoretical analysis is studied via the Lyapunov stability theory to prove the asymptotic stability of the closed-loop system. The superiority and effectiveness of the proposed method are demonstrated with comparative numerical simulation and experiment with a robotic test-rig.