Maintaining the integrity of their protective plasma membrane is a primary requirement of cells. Accordingly, cellular events that breach the membrane are tightly regulated. Artificial vesicles used in drug delivery must also stay intact until they have reached the desired target. In both cases, the intrinsic resistance of the membrane to rupture must be overcome to allow the efflux of the vesicle's contents. Here, we use mesoscopic simulations to study the fusion of 28-nm-diameter vesicles to 50 x 50 nm(2) planar membrane patches over 2 mus. We monitor the time evolution of 93 different fusion attempts. This allows us to construct a global morphology diagram, using the initial tensions of the vesicle and the planar membrane patch as control parameters, and to determine the corresponding fusion statistics. All successful fusion events are observed to occur within 350 ns, which reflects the presence of alternative pathways for the tension relaxation.