In this paper, the regularity and finite difference methods for the two-dimensional delay fractional equations are considered. The analytic solution is derived by eigenvalue expansions and Laplace transformation. However, due to the derivative discontinuities resulting from the delay effect, the traditional L1-ADI scheme fails to achieve the optimal convergence order. To overcome this issue and improve the convergence order, a simple and cost-effective decomposition technique is introduced and a fitted L1-ADI scheme is proposed. The numerical tests are conducted to verify the theoretical result.