We develop a variational approach to simulating the dynamics of open quantum many-body systems using deep autoregressive neural networks. The parameters of a compressed representation of a mixed quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. We illustrate our approach by solving the dissipative quantum Heisenberg model in one dimension for up to 40 spins and in two dimensions for a 4×4 system and by applying it to the simulation of confinement dynamics in the presence of dissipation.