We propose a novel feature-centric approach to surrogate modeling of dynamical systems driven by time-varying exogenous excitations. This approach, named Functional Nonlinear AutoRegressive with eXogenous inputs (F-NARX), aims to approximate the system response based on temporal features of both the exogenous inputs and the system response, rather than on their values at specific time lags. This is a major step away from the discrete-time-centric approach of classical NARX models, which attempts to determine the relationship between selected time steps of the input/output time series. By modeling the system in a time-feature space instead of the original time axis, F-NARX can provide more stable long-term predictions and drastically reduce the reliance of the model performance on the time discretization of the problem. F-NARX, like NARX, acts as a framework and is not tied to a single implementation. In this work, we introduce an F-NARX implementation based on principal component analysis and polynomial basis functions. To further improve prediction accuracy and computational efficiency, we also introduce a strategy to identify and fit a sparse model structure, thanks to a modified hybrid least angle regression approach that minimizes the expected forecast error, rather than the one-step-ahead prediction error. Since F-NARX is particularly well-suited to modeling engineering structures typically governed by physical processes, we investigate the behavior and capabilities of our F-NARX implementation on two case studies: an eight-story building under wind loading and a threestory steel frame under seismic loading. While the first case study highlights the simple and intuitive parametrization of the presented F-NARX implementation, the second case study demonstrates its high accuracy and prediction stability in more complex nonlinear problems. Our results demonstrate that F-NARX has several favorable properties over classical NARX, making it well suited to emulate engineering systems with high accuracy over extended time periods.