We present the results of simulations in two spatial dimensions of the very early phases of the explosion of SN 1987A. Starting our calculations close to the time when the prompt shock emerges from the neutrino-sphere, we simulate the first few seconds of the explosion using a two-dimensional cylindrical geometry smooth particle hydrodynamics (SPH) code. The explosion energy is obtained, as in the delayed mechanism, by tapping a small fraction of the energy carried by the neutrinos produced in the making of the neutron star. The success of the explosion is determined to be sensitive to the duration of the infall, the timing of the bounce, and the subsequent neutrino heating. Using a semianalytical model for the initial structure of the collapsed object, we present two simulations that differ by the mass that has been allowed to collapse into a neutron star prior to the bounce. In the case of a short initial infall (1.37 M. baryon mass proto-neutron star), the explosion fails due to excessive cooling. For a longer initial infall (1.5 M. baryon mass proto-neutron star), the cooling is less and the explosion is successful although relatively weak [approximately 0.35 foe (10(51) ergs)]. We show that in this case, a successful explosion is brought about by the presence of an entropy gradient which, combined with the gravitational pull of the neutron star, leads to extremely strong instabilities. In such models, convective currents carry the deposited energy away from the neutron star and bring in new, cold material, thus preventing excessive neutrino cooling. The flow is coherent on very large scales and does not lead to microscopic mixing. The critical importance of the global circulation, for the success of the explosion, is demonstrated by the fact that similar simulations, carried out with a one-dimensional code, fizzle dramatically. It also appears that the early mixing during the explosion could account for a number of observational features of SN 1987A, such as infrared iron line profiles with extended high-velocity wings, which have so far remained unexplained.