According to recent observations the topological surface states of Weyl semimetals, their Fermi arcs, may develop a superconducting gap, while bulk superconductivity remains absent. What drives the formation of this novel superconducting state is an open question. Here we show that surface superconductivity in time-reversal-invariant Weyl semimetals can arise from the interaction of Fermi arc electrons with both surface and bulk phonons. Employing a low-energy Hamiltonian including Coulomb repulsion and electron-phonon interactions, we derive the effective theory on the Fermi arcs. We identify two competing pairing channels, intra-arc and inter-arc, whose relative strength is governed by the efficiency of Coulomb screening at the surface. The combined effect of the Fermi arcs being disconnected and weak screening of the Coulomb repulsion at the system’s surface causes nodes to appear in the superconducting gap, as observed recently by photoelectron spectroscopy experiments on PtBi2. This suggests manipulation of the Coulomb screening, e.g. by a surface layer coating, as a pathway to engineer the critical temperature, as well as size and even symmetry of the surface superconducting gap.