In this study, an innovative technique to design an observer-based finite-time output feedback controller (FT-OFC) is proposed for a class of non-linear systems. This controller aims to make the state variables converge to a small bound around the origin in a finite time. The main innovation of this study is to transform the non-linear system into a new time-varying form to achieve the finite-time boundedness criteria using the asymptotic stability methods. Moreover, without any prior knowledge of the upper bounds of the system uncertainties and/or disturbances, and only based on the output measurements, a novel timevarying extended state observer is designed to estimate the states of the non-linear system as well as the uncertainties and disturbances in a finite time. In this way, the time-varying gains of the extended state observer are designed to converge the observation error to a neighbourhood of zero while remaining uniformly bounded in finite time. Subsequently, an observer-based time-varying control law is designed to make the system globally uniformly bounded in finite time. Finally, the efficiency of the proposed FT-OFC for a disturbed double integrator system with unknown measurement noise is illustrated by numerical simulations.