Nuclear star clusters around a central massive black hole (MBH) are expected to be abundant in stellar black hole (BH) remnants and BH-BH binaries. These binaries form a hierarchical triple system with the central MBH, and gravitational perturbations from the MBH can cause high-eccentricity excitation in the BH-BH binary orbit. During this process, the eccentricity may approach unity, and the pericenter distance may become sufficiently small so that gravitational-wave emission drives the BH-BH binary to merge. In this work, we construct asimple proof-of-concept model for this process, and specifically, we study the eccentric Kozai-Lidov mechanism in unequal-mass, soft BH-BHbinaries. Our model is based on a set of Monte Carlo simulations for BH-BH binaries in galactic nuclei, taking into account quadrupole- and octupole-level secular perturbations, general relativistic precession,and gravitational-wave emission. For a typical steady-state number of BH-BH binaries, our model predicts a total merger rate of ̃1-3 {Gpc}^-3 {yr} ^-1, depending on the assumed density profile in the nucleus. Thus, our mechanism could potentially compete with other dynamical formation processes for merging BH-BH binaries,such as the interactions of stellar BHs in globular clusters or in nuclear star clusters without an MBH.