We establish a localization principle for odd-dimensional supergravity theories containing Chern-Simons interactions. Our construction relies on combining an equivariant extension of the gauge-invariant part of the action with an equivariant completion of the Chern-Simons anomaly form ${\Phi }^{\mathrm{\text{(anom)}}}$ to form a relative equivariant cohomology class. This leads to a universal fixed point formula for the on-shell action of supersymmetric solutions that is expressed in terms of ${\Phi }^{\mathrm{\text{(anom)}}}$ and an equivariant Euler class. We illustrate the formalism in $D=11$ and $D=5$ supergravity, with applications including the supersymmetric Casimir energy and superconformal index of M5-branes, as well as black ring and black lens solutions.