Retrieval-augmented generation models have shown state-of-the-art performance across many knowledge-intensive NLP tasks such as open question answering and fact verification. These models are trained to generate the final output given the retrieved passages, which can be irrelevant to the original query, leading to learning spurious cues or answer memorization. This work introduces a method to incorporate the evidentiality of passages -- whether a passage contains correct evidence to support the output -- into training the generator. We introduce a multi-task learning framework to jointly generate the final output and predict the evidentiality of each passage, leveraging a new task-agnostic method to obtain silver evidentiality labels for supervision. Our experiments on five datasets across three knowledge-intensive tasks show that our new evidentiality-guided generator significantly outperforms its direct counterpart with the same-size model and advances the state of the art on FaVIQ-Ambig. We attribute these improvements to both the auxiliary multi-task learning and silver evidentiality mining techniques.