Several experimental platforms, such as superconducting circuits or ultracold atomic in optical lattices, nowadays allow to probe many-body physics in unprecedented regimes, such as in non-equilibrium conditions resulting from controlled dissipation and driving, but theoretical techniques for describing those regimes are limited. In this work , we introduce an extension of the nonequilibrium dynamical mean-field theory (DMFT) for bosonic lattice models described by Markovian master equations. DMFT maps these lattice problems onto a problem of a single site coupled to a classical field and to a non-interacting bath, accounting for leading corrections to Gutzwiller mean-field theory due to finite dimensionality. Our approach relies on a new method for solving the effective single-site problem based on a non-crossing approximation in the coupling to the DMFT bath, going beyond standard Born-Markov approximations . We then discuss an application to a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump, computing its steady-state properties. DMFT captures hopping-induced processes that are completely missed by Gutzwiller mean-field theory, which are crucial to obtain the correct stationary-state, such as to describe its quantum-Zeno behaviour when the losses are strong, or to predict the critical hopping for a phase transition between an incoherent phase and a coherent, limit-cycle phase.  O. Scarlatella, A. A. Clerk, R. Fazio, and M. Schiró, Dynamical Mean-Field Theory for Markovian Open Quantum Many-Body Systems, Phys. Rev. X 11, 031018 (2021).  O. Scarlatella and M. Schiro, Self-Consistent Dynamical Maps for Open Quantum Systems, arXiv:2107.05553.