Semantic segmentation is a challenging task in the absence of densely labelled data. Only relying on class activation maps (CAM) with image-level labels provides deficient segmentation supervision. Prior works thus consider pre-trained models to produce coarse saliency maps to guide the generation of pseudo segmentation labels. However, the commonly used off-line heuristic generation process cannot fully exploit the benefits of these coarse saliency maps. Motivated by the significant inter-task correlation, we propose a novel weakly supervised multi-task framework termed as AuxSegNet, to leverage saliency detection and multi-label image classification as auxiliary tasks to improve the primary task of semantic segmentation using only image-level ground-truth labels. Inspired by their similar structured semantics, we also propose to learn a cross-task global pixel-level affinity map from the saliency and segmentation representations. The learned cross-task affinity can be used to refine saliency predictions and propagate CAM maps to provide improved pseudo labels for both tasks. The mutual boost between pseudo label updating and cross-task affinity learning enables iterative improvements on segmentation performance. Extensive experiments demonstrate the effectiveness of the proposed auxiliary learning network structure and the cross-task affinity learning method. The proposed approach achieves state-of-the-art weakly supervised segmentation performance on the challenging PASCAL VOC 2012 and MS COCO benchmarks.