The well-known prescribed-time stabilizing design faces the need of using time-varying high gains which escape to infinity as time approaches the desired instant. In the presence of external disturbances and measurement noise, the corresponding state response is significantly amplified as well that leads to the lack of robustness in the closed-loop implementation. For eliminating this drawback, the implicit Euler discretization was recently developed for state feedback and observer design. The resulting implementation conserved desired robustness properties beyond the prescribed-time interval and also ensured bounded state dynamics in the presence of measurement noise. In the present work, the proposed robust prescribed-time design is further developed towards its output feedback extension the separation principle is not valid in this setting). The effectiveness of the design is additionally supported by numerical experiments.