Dynamical models are often corrupted by model uncertainties, external disturbances, and measurement noise. These factors affect the performance of model-based observers and as a result, affect the closed-loop performance. Therefore, it is critical to develop robust model-based estimators that reconstruct both the states and the model disturbances while mitigating the effect of measurement noise in order to ensure good system monitoring and closed-loop performance when designing controllers. In this article, a robust step by step non-asymptotic observer for triangular nonlinear systems for the joint estimation of the state and the disturbance is developed. The proposed approach provides a sequential estimation of the states and the disturbance in finite time using smooth modulating functions. The robustness of the proposed observer is both in the sense of model disturbances and measurement noise. In fact, the structure of triangular systems combined with the modulating function-based method allows the estimation of the states independently of model disturbances and the integral operator involved in the modulating function-based method mitigates the noise. Additionally, the modulating function method shifts the derivative from the noisy output to the smooth modulating function which strengthens its robustness properties. The applicability of the proposed modulating function-based estimator is illustrated in numerical simulations and compared to a second-order sliding mode super twisting observer under different measurement noise levels.