We identify two fundamental properties of the Gor'kov Green's function of finite one-dimensional topological superconductors. In the low-frequency (low-energy) regime, the normal and anomalous Green's functions become identical up to a phase factor. Moreover, they exhibit pronounced nonlocality: the components connecting the two ends of the system grow exponentially with system length, whereas the local components at either end vanish in the zero-frequency limit. These striking features signify the emergence of unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode composed of hybridized Majorana end modes. The nonlocal Cooper pairs are directly linked to fermion parity and to the nonlocal transport properties of finite topological superconductors. By uncovering the unusual nonlocal properties of the Gor'kov Green's function, our analysis advances the understanding of Majorana nonlocality, a key concept in topological quantum computation.