We study the problem of estimating the origin of an epidemic outbreak: given a contact network and a snapshot of epidemic spread at a certain time, determine the infection source. This problem is important in different contexts of computer or social networks. Assuming that the epidemic spread follows the usual susceptible-infected-recovered model, we introduce an inference algorithm based on dynamic message-passing equations and we show that it leads to significant improvement of performance compared to existing approaches. Importantly, this algorithm remains efficient in the case where the snapshot sees only a part of the network.