Summary: In this paper, we study the following critical system with fractional Laplacian: \[ \begin{cases} (-\Delta)^su+\lambda_1u=\mu_1|u|^{2^{\ast}-2}u+\frac{\alpha \gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \& \text{in } \Omega, \\ (-\Delta)^sv+\lambda_2v= \mu_2|v|^{2^{\ast}-2}v+\frac{\beta \gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v \& \text{in } \Omega, \\ u=v=0 & \text{in } \mathbb{R}^N\setminus\Omega, \end{cases} \] where \((-\Delta)^s\) is the fractional Laplacian, \(0< s<1\), \(\mu_1,\mu_2>0\), \(2^{\ast}=\frac{2N}{N-2s}\) is a fractional critical Sobolev exponent, \(N>2s, 1<\alpha, \beta<2, \alpha+\beta=2^{\ast} , \Omega\) is an open bounded set of \(\mathbb{R}^N\) with Lipschitz boundary and \(\lambda_1,\lambda_2>-\lambda_{1,s}(\Omega), \lambda_{1,s}(\Omega)\) is the first eigenvalue of the non-local operator \((-\Delta)^s\) with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all \(\gamma>0\). Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when \(\gamma\rightarrow 0\).