We construct, to leading orders in the momentum expansion, an effective theory of a chiral (px+ipy) two-dimensional fermionic superfluid at zero temperature that is consistent with nonrelativistic general coordinate invariance. This theory naturally incorporates the parity and time-reversal violating effects such as the Hall viscosity and the edge current. The particle number current and stress tensor are computed and their linear response to electromagnetic and gravitational sources is calculated. We also consider an isolated vortex in a chiral superfluid and identify the leading chirality effect in the density depletion profile.