We present a transient absorption setup combining broadband detection over the visible-UV range with high temporal resolution (similar to 20 fs) which is ideally suited to trigger and detect vibrational coherences in different classes of materials. We generate and detect coherent phonons (CPs) in single-layer (1L)-MoS2, as a representative semiconducting 1L-transition metal dichalcogenide (TMD), where the confined dynamical interaction between excitons and phonons is unexplored. The coherent oscillatory motion of the out-of-plane A'(1) phonons, triggered by the ultrashort laser pulses, dynamically modulates the excitonic resonances on a time scale of few tens of fs. We observe an enhancement by almost 2 orders of magnitude of the CP amplitude when detected in resonance with the C exciton peak, combined with a resonant enhancement of CP generation efficiency. Ab initio calculations of the change in the 1L-MoS2 band structure induced by the A'(1) phonon displacement confirm a strong coupling with the C exciton. The resonant behavior of the CP amplitude follows the same spectral profile of the calculated Raman susceptibility tensor. These results explain the CP generation process in 1L-TMDs and demonstrate that CP excitation in 1L-MoS2 can be described as a Raman-like scattering process.