Computational drug target prediction has become an important process in drug discovery. Network-based approaches are commonly used in computational drug-target interaction (DTI) prediction. Existing network-based approaches are limited in capturing the contextual information on how diseases, drugs and genes are connected. Here, we proposed a context-sensitive network (CSN) model for DTI prediction by modeling contextual drug phenotypic relationships. We constructed a Drug-Side Effect Context-Sensitive Network (DSE-CSN) of 139 760 drug-side effect pairs, representing 1480 drugs and 5868 side effects. We also built a protein-protein interaction network (PPIN) of 15 267 gene nodes and 178 972 weighted edges. A heterogeneous network was built by connecting the DSE-CSN and the PPIN through 3684 known DTIs. For each drug on the DSE-CSN, its genetic targets were predicted and prioritized using a network-based ranking algorithm. Our approach was evaluated in both de novo and leave-one-out cross-validation analysis using known DTIs as the gold standard. We compared our DSE-CSN-based model to the traditional similarity-based network (SBN)-based prediction model. The results suggested that the DSE-CSN-based model was able to rank known DTIs highly. In a de novo cross-validation, the area under the receiver operating characteristic (ROC) curve was 0.95. In a leave-one-out cross-validation, the average rank was top 3.2% for known DTIs. When it was compared to the SBN-based model using the Precision-Recall curve, our CSN-based model achieved a higher mean average precision (MAP) (0.23 versus 0.19, P-value<1e-4) in a de novo cross-validation analysis. We further improved the CSN-based DTI prediction by differentially weighting the drug-side effect pairs on the network and showed a significant improvement of the MAP (0.29 versus 0.23, P-value<1e-4). We also showed that the CSN-based model consistently achieved better performances than the traditional SBN-based model across different drug classes. Moreover, we demonstrated that our novel DTI predictions can be supported by published literature. In summary, the CSN-based model, by modeling the context-specific inter-relationships among drugs and side effects, has a high potential in drug target prediction.