  The electronic structure, lattice dynamics, and electron-phonon coupling
(EPC) of the newly discovered (Ba,K)SbO$_3$ superconductors are investigated by
first-principles calculations. The EPC of (Ba,K)SbO$_3$ is significantly
enhanced by considering non-local electronic correlation using the
Heyd-Scuseria-Ernzerhof hybrid exchange-correlation functional (HSE06). The EPC
strength {\lambda} of Ba$_{0.35}$K$_{0.65}$SbO$_3$ is strongly increased from
0.33 in local-density approximation calculations to 0.59 in HSE06 calculations,
resulting in a superconducting transition temperature Tc of about 14.9 K, which
is in excellent agreement with experimental value of ~ 15 K. Our findings
suggest (Ba,K)SbO$_3$ are extraordinary conventional superconductors, where
non-local electronic correlation expands the bandwidth, enhances the EPC, and
boosts the Tc. Moreover, we find both {\lambda} and Tc depend crucially on the
K-doping level for (Ba,K)SbO$_3$ and (Ba,K)SbO$_3$ compounds. (Ba,K)SbO$_3$
have stronger EPC strength and higher Tc than those of (Ba,K)SbO$_3$ at the
same K-doping level.
