The island formula—an extremization prescription for generalized entropy—is known to result in a unitary Page curve for the entropy of Hawking radiation. This semiclassical entropy formula has been derived for Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the “replica trick” to evaluate the Euclidean path integral. Alternatively, for eternal Anti–de Sitter black holes, we derive the extremization of generalized entropy from minimizing the microcanonical action of an entanglement wedge. The on-shell action is minus the entropy and arises in the saddle-point approximation of the (nonreplicated) microcanonical path integral. When the black hole is coupled to a bath, islands emerge from maximizing the entropy at fixed energy, consistent with the island formula. Our method applies to JT gravity as well as other two-dimensional dilaton gravity theories.