N-body simulations predict that dark matter haloes are described by specific density profiles on both galactic- and cluster-sized scales. Weak gravitational lensing through the measurements of their first and second order properties, shear and flexion, is a powerful observational tool for investigating the true shape of these profiles. One of the three-parameter density profiles recently favoured in the description of dark matter haloes is the Einasto profile. We present exact expressions for the shear and the first and second flexions of Einasto dark matter haloes derived using a Mellin-transform formalism in terms of the Fox H and Meijer G functions, that are valid for general values of the Einasto index. The resulting expressions can be written as series expansions that permit us to investigate the asymptotic behaviour of these quantities. Moreover, we compare the shear and flexion of the Einasto profile with those of different mass profiles including the singular isothermal sphere, the Navarro-Frenk-White profile, and the Sérsic profile. We investigate the concentration and index dependences of the Einasto profile, finding that the shear and second flexion could be used to determine the halo concentration, whilst for the Einasto index the shear and first and second flexions may be employed. We also provide simplified expressions for the weak lensing properties and other lensing quantities in terms of the generalized hypergeometric function. ; Comment: 14 pages, 3 figures. Accepted for publication in Astronomy and Astrophysics