Electron-phonon coupling is generally expected to suppress band topology, driving a topological band insulator into a trivial phase. Here we show that, using the prototypical Haldane-Holstein model at partial filling of a topological band, strong electron-phonon coupling can instead stabilize topological phases of matter. The electron-phonon coupling plays a double role: it generates the longer-range interactions that correlate the carriers and, in the attractive channel, also binds them into bosonic pairs. For spinful electrons, it pairs opposite spins into charge-2e bipolarons that form a bosonic fractional Chern insulator (FCI) at filling ν=1/2 with charge-e semionic excitations. Tuning the band topology and the coupling strength maps out a rich phase diagram containing this bosonic FCI, superconductors condensed at the M and K points, and several charge-ordered solids. For spin-polarized electrons, strong coupling instead stabilizes a C=2 quantum anomalous Hall crystal (QAHC) at ν=1/3. Electron-phonon coupling thus emerges as a route to, rather than an obstruction against, topology in partially filled Chern bands.