We calculate the nucleon electric dipole moment (EDM) from the $\theta$ term with overlap fermions on three domain wall lattices with different sea pion masses at lattice spacing 0.11 fm. Due to the chiral symmetry conserved by the overlap fermions, we have well defined topological charge and chiral limit for the EDM. Thus, the chiral extrapolation can be carried out reliably at nonzero lattice spacings. We use three to four different partially quenched valence pion masses for each sea pion mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely, which can be described by partially quenched chiral perturbation theory. With the help of the cluster decomposition error reduction (CDER) technique, we determine the neutron and proton EDM at the physical pion mass to be $d_{n}=-0.00148\left(14\right)\left(31\right)\bar\theta$ e$\cdot$fm and $d_{p}=0.0038\left(11\right)\left(8\right)\bar\theta$ e$\cdot$fm. This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM calculation and paves the road to future precise studies of the strong $CP$ violation effects.