The tunneling approach for entropy generation in quantum gravity is shown to be valid when applied to three-dimensional (3D) general relativity. The entropy of de Sitter and Reissner-Nordström external event horizons and of the 3D black hole obtained by Bañados and co-workers is rederived from the tunneling of the metric to these spacetimes. The analysis for spacetimes with an external horizon is carried out in complete analogy with the 4D case. However, we find significant differences for the black hole. It was previously shown that the tunneling approach implies a large, maybe infinite, degeneracy of the configuration that tunnels to the four-dimensional black hole. For the three-dimensional black hole we find that the initial configuration may not yield such a highly degenerate object.