Peritectic crystallization is a process in which the solid phase precipitated in the form of solid solution reacts with the liquid phase to form another solid phase. The process can be described by a phase field model where two continuous phase variables, $\phi$ and $\psi$, are introduced to distinguish the three different phases. We discretize the time variable with a scalar auxiliary variable (SAV) method that can ensure the unconditional energy stability. Moreover, the SAV method only requires solving a linear system at each time step and therefore reduces the computational complexity. The space variables in a two-dimensional region are discretized by an operator splitting method equipped with a high order compact finite difference formulation. This approach is effective and convenient since only a series of one-dimensional problems need to be solved at each step. We prove the unconditional energy stability theoretically and test the order of convergence and energy stability through numerical experiments. Simulations of peritectic solidification demonstrate the patterns formed during the process.