Two quasi-subordination subclasses \(\mathcal{Q}\Sigma^{\gamma,k}_{ \alpha,\beta}(\vartheta,\rho;\phi)\) and \(\mathcal{M}\Sigma^{\gamma,k}_{ \alpha,\beta}(\tau,\vartheta,\rho;\phi)\) of the class \(\Sigma\) of analytic and bi-univalent functions associated with the convolution operator involving Mittag-Leffler function are introduced and investigated. Then, the corresponding bound estimates of the coefficients \(a_2\) and \(a_3\) are provided. Meanwhile, Fekete-Szegö functional inequalities for these classes are proved. Besides, some consequences and connections to all the theorems would be interpreted, which generalize and improve earlier known results.