It is well-known that the only minimal Lagrangian submanifolds of constant sectional curvature $c$ in a Riemannian complex space form of constant holomorphic sectional curvature $4c$ are the totally geodesic ones. In this paper we investigate minimal Lagrangian Lorentzian submanifolds of constant sectional curvature $c$ in Lorentzian complex space form of constant holomorphic sectional curvature $4c$. We prove that the situation in the Lorentzian case is quite different from the Riemannian case. Several existence and classification theorems in this respect are obtained. Some explicit expression of flat minimal Lagrangian submanifolds in flat complex Lorentzian space form are also presented.