Open-set domain adaptation aims to improve the generalization performance of a learning algorithm on a more realistic problem of open-set domain shift where the target data contains an additional unknown class that is not present in the source data. Most existing algorithms include two phases that can be described as closed-set domain adaptation given heuristic unknown class separation. Therefore, the generalization error cannot be strictly bounded due to the gap between the true distribution and samples inferred from heuristics. In this paper, we propose an end-to-end algorithm that tightly bound the risk of the entire target task by positive-unlabeled (PU) learning theory and the joint error from domain adaptation. Extensive experiments on various data sets demonstrate the effectiveness and efficiency of our proposed algorithm over open-set domain adaptation baselines.