We present an efficient computational method to obtain accurate values for energy positions and widths of autoionizing states in Be-like atomic systems. The two-active (outer) electron wavefunction is expanded in terms of Hylleraas-type correlated configurations. The interaction with the 1s2 frozen core is represented through a model potential and the unphysical 1snℓ series of virtual core states are removed by using a Phillips-Kleinman pseudopotential projector. A novel feature is that all matrix elements can be written in closed form. We illustrate the performance of our approach in computing doubly-excited states in Be and Ne6+ by using the stabilization method.