In this paper, $k$-fractional integral operators containing further extension of Mittag-Leffler function are defined firstly. Then, the first and second version of Hadamard and Fejér-Hadamard inequalities for generalized $k$-fractional integrals are obtained. Finally, by using these generalized $k$-fractional integrals containing Mittag-Leffler functions, results for $p$-convex functions are obtained. The results for convex functions can be deduced by taking $p=1$.