Targeting at recognizing and localizing action instances with only video-level labels during training, Weakly-supervised Temporal Action Localization (WTAL) has achieved significant progress in recent years. However, living in the dynamically changing open world where unknown actions constantly spring up, the closed-set assumption of existing WTAL methods is invalid. Compared with traditional open-set recognition tasks, Open-world WTAL (OWTAL) is challenging since not only are the annotations of unknown samples unavailable, but also the fine-grained annotations of known action instances can only be inferred ambiguously from the video category labels. To address this problem, we propose a Cascade Evidential Learning framework at an evidence level, which targets at OWTAL for the first time. Our method jointly leverages multi-scale temporal contexts and knowledge-guided prototype information to progressively collect cascade and enhanced evidence for known action, unknown action, and background separation. Extensive experiments conducted on THUMOS-14 and ActivityNet-v1.3 verify the effectiveness of our method. Besides the classification metrics adopted by previous open-set recognition methods, we also evaluate our method on localization metrics which are more reasonable for OWTAL.