In neural machine translation (NMT), monolingual data are usually exploited through a so-called back-translation: sentences in the target language are translated into the source language to synthesize new parallel data. While this method provides more training data to better model the target language, on the source side, it only exploits translations that the NMT system is already able to generate using a model trained on existing parallel data. In this work, we assume that new translation knowledge can be extracted from monolingual data, without relying at all on existing parallel data. We propose a new algorithm for extracting from monolingual data what we call partial translations: pairs of source and target sentences that contain sequences of tokens that are translations of each other. Our algorithm is fully unsupervised and takes only source and target monolingual data as input. Our empirical evaluation points out that our partial translations can be used in combination with back-translation to further improve NMT models. Furthermore, while partial translations are particularly useful for low-resource language pairs, they can also be successfully exploited in resource-rich scenarios to improve translation quality.