In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution $u$ and the flux $\sigma$ are approximated using finite element spaces consisting of piecewise polynomials of degree $k$ and $r$ respectively. Based on interpolation operators and an auxiliary projection, superconvergent $H^1-$error estimates of both the primary solution approximation $u_h$ and the flux approximation $\sigma_h$ are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of $O(h^{r+2})$ for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order $r$ are employed with optimal error estimate of $O(h^{r+1})$.