We present comboFM, a machine learning framework for predicting the responses of drug combinations in pre-clinical studies, such as those based on cell lines or patient-derived cells. comboFM models the cell context-specific drug interactions through higher-order tensors, and efficiently learns latent factors of the tensor using powerful factorization machines. The approach enables comboFM to leverage information from previous experiments performed on similar drugs and cells when predicting responses of new combinations in so far untested cells; thereby, it achieves highly accurate predictions despite sparsely populated data tensors. We demonstrate high predictive performance of comboFM in various prediction scenarios using data from cancer cell line pharmacogenomic screens. Subsequent experimental validation of a set of previously untested drug combinations further supports the practical and robust applicability of comboFM. For instance, we confirm a novel synergy between anaplastic lymphoma kinase (ALK) inhibitor crizotinib and proteasome inhibitor bortezomib in lymphoma cells. Overall, our results demonstrate that comboFM provides an effective means for systematic pre-screening of drug combinations to support precision oncology applications.