Preclinical models have been the workhorse of cancer research, producing massive amounts of drug response data. Unfortunately, translating response biomarkers derived from these datasets to human tumors has proven to be particularly challenging. To address this challenge, we developed TRANSACT, a computational framework that builds a consensus space to capture biological processes common to preclinical models and human tumors and exploits this space to construct drug response predictors that robustly transfer from preclinical models to human tumors. TRANSACT performs favorably compared to four competing approaches, including two deep learning approaches, on a set of 23 drug prediction challenges on The Cancer Genome Atlas and 226 metastatic tumors from the Hartwig Medical Foundation. We demonstrate that response predictions deliver a robust performance for a number of therapies of high clinical importance: platinum-based chemotherapies, gemcitabine, and paclitaxel. In contrast to other approaches, we demonstrate the interpretability of the TRANSACT predictors by correctly identifying known biomarkers of targeted therapies, and we propose potential mechanisms that mediate the resistance to two chemotherapeutic agents.