Deep learning brought boosts to auto diabetic retinopathy (DR) diagnosis, thus, greatly helping ophthalmologists for early disease detection, which contributes to preventing disease deterioration that may eventually lead to blindness. It has been proved that convolutional neural network (CNN)-aided lesion identifying or segmentation benefits auto DR screening. The key to fine-grained lesion tasks mainly lies in: (1) extracting features being both sensitive to tiny lesions and robust against DR-irrelevant interference, and (2) exploiting and re-using encoded information to restore lesion locations under extremely imbalanced data distribution. To this end, we propose a CNN-based DR diagnosis network with attention mechanism involved, termed lesion-aware network, to better capture lesion information from imbalanced data. Specifically, we design the lesion-aware module (LAM) to capture noise-like lesion areas across deeper layers, and the feature-preserve module (FPM) to assist shallow-to-deep feature fusion. Afterward, the proposed lesion-aware network (LANet) is constructed by embedding the LAM and FPM into the CNN decoders for DR-related information utilization. The proposed LANet is then further extended to a DR screening network by adding a classification layer. Through experiments on three public fundus datasets with pixel-level annotations, our method outperforms the mainstream methods with an area under curve of 0.967 in DR screening, and increases the overall average precision by 7.6%, 2.1%, and 1.2% in lesion segmentation on three datasets. Besides, the ablation study validates the effectiveness of the proposed sub-modules.