The mechanism of fluctuational transitions between coexisting chaotic attractors (CA) separated by a fractal basin boundary (FBB) was studied. A two-dimensional map with local disconnected FBBs was modeled numerically. After excitation with a weak noise, the escape trajectories between the CAs and the noise realizations that induced them were collected. The unique most probable escape path (MPEP) from the CA to the FBB was determined using statistical analyses of fluctuational trajectories and the Hamiltonian fluctuation theory.