Automated fine-grained diabetic retinopathy (DR) grading was of great significance for assisting ophthalmologists in monitoring DR and designing tailored treatments for patients. Nevertheless, it is a challenging task as a result of high intra-class variations, high inter-class similarities, small lesions, and imbalanced data distributions. The pivotal factor for the success in fine-grained DR grading is to discern more subtle associated lesion features, such as microaneurysms (MA), Hemorrhages (HM), soft exudates (SE), and hard exudates (HE). In this paper, we constructed a simple yet effective deep attentive convolutional neural network (DACNN) for DR grading and lesion discovery with only image-wise supervision. Designed as a top-down architecture, our model incorporated stochastic atrous spatial pyramid pooling (sASPP), global attention mechanism (GAM), category attention mechanism (CAM), and learnable connected module (LCM) to better extract lesion-related features and maximize the DR grading performance. To be concrete, we devised sASPP combining randomness with atrous spatial pyramid pooling (ASPP) to accommodate the various scales of the lesions and struggle against the co-adaptation of multiple atrous convolutions. Then, GAM was introduced to extract class-agnostic global attention feature details, whilst CAM was explored for seeking class-specific distinctive region-level lesion feature information and regarding each DR severity grade in an equal way, which tackled the problem of imbalance DR data distributions. Further, the LCM was designed to automatically and adaptively search the optimal connections among layers for better extracting detailed small lesion feature representations. The proposed approach obtained high accuracy of 88.0% and kappa score of 88.6% for multi-class DR grading task on the EyePACS dataset, respectively, while 98.5% AUC, 93.8% accuracy, 87.9% kappa, 90.7% recall, 94.6% precision, and 92.6% F1-score for referral and non-referral classification on the Messidor dataset. Extensive experimental results on three challenging benchmarks demonstrated that the proposed approach achieved competitive performance in DR grading and lesion discovery using retinal fundus images compared with existing cutting-edge methods, and had good generalization capacity for unseen DR datasets. These promising results highlighted its potential as an efficient and reliable tool to assist ophthalmologists in large-scale DR screening.