We present an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. We focus on elliptic Lagrangian surfaces, a spacelike property inferred from the identification of the Lagrangian Grassmannian with the Einstein universe. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the questions of symplectic applicability and deformation for elliptic Lagrangian immersions.