River flow plays a vital role in the hydrologic cycle. In recent years, numerous machine learning approaches, particularly deep neural networks (DNNs), have demonstrated success in forecasting river flow. However, many of these models depend heavily on extensive historical flow data and very specific hydrological features, which are often labor-intensive to gather. Moreover, most existing predictive models, especially DNN-based, are handcrafted, requiring substantial domain expertise and extensive hyperparameter tuning, making the modeling process potentially inefficient. In this study, we propose a novel approach to achieve flexible and efficient river flow prediction that relies solely on openly accessible weather forecast data. A one-dimensional convolutional neural network (1D-CNN) is designed to effectively capture the temporal relationships between weather variables and river flow. Additionally, an efficient evolutionary neural architecture search (NAS) algorithm is developed to automatically discover the best 1D-CNN architectures, thereby improving predictive performance while reducing the need for manual architecture tuning. To evaluate our approach (named AutoNN-Flow), experiments are conducted using the Kaeo River in New Zealand as a case study. Without access to river-specific attributes, AutoNN-Flow achieves highly accurate 7-day and 14-day river flow predictions, greatly outperforming classic machine learning models.