We generalize the numerical linked-cluster expansion (NLCE) method to study the dissipative quantum many-body system. We apply the NLCE to the triangular strip and two-dimensional lattice system. We investigate the dynamics and steady-state properties of the dissipative XYZ model where the coherent dynamics is governed by the anisotropic Heisenberg Hamiltonian while the nonunitary process is induced by the incoherent spin flips. By comparing with the quantum trajectory simulations, the NLCE results show good performance in capturing the dynamics of system with short-range correlations. For strong and long-range correlated system, the larger size clusters in the series should be included. The NLCE study for the magnetic susceptibility also signals the steady-state paramagnetic-ferromagnetic phase transition in the two-dimensional case.