Image segmentation models trained only with image-level labels have become increasingly popular as they require significantly less annotation effort than models trained with scribble, bounding box or pixel-wise annotations. While methods utilizing image-level labels achieve promising performance for the segmentation of larger-scale objects, they perform less well for the fine structures frequently encountered in biological images. In order to address this performance gap, we propose a deep network architecture based on two key principles, Global Weighted Pooling (GWP) and segmentation refinement by low-level image cues, that, together, make segmentation of fine structures possible. We apply our segmentation method to image datasets containing such fine structures, nematodes (worms + eggs) and nematode cysts immersed in organic debris objects, which is an application scenario encountered in automated soil sample screening. Supervised only with image-level labels, our approach achieves Dice coefficients of 79.72% and 58.51 % for nematode and nematode cyst segmentation, respectively.