The growing importance of intraday electricity trading in Europe calls for improved price forecasting and tailored decision-support tools. In this paper, we propose a novel generative neural network model to generate probabilistic path forecasts for intraday electricity prices and use them to construct effective trading strategies for Germany's continuous-time intraday market. Our method demonstrates competitive performance in terms of statistical evaluation metrics compared to two state-of-the-art statistical benchmark approaches. To further assess its economic value, we consider a realistic fixed-volume trading scenario and propose various strategies for placing market sell orders based on the path forecasts. Among the different trading strategies, the price paths generated by our generative model lead to higher profit gains than the benchmark methods. Our findings highlight the potential of generative machine learning tools in electricity price forecasting and underscore the importance of economic evaluation.