We report the measurement of sub-MeV solar neutrinos through the use of their associated Cherenkov radiation, performed with the Borexino detector at the Laboratori Nazionali del Gran Sasso. The measurement is achieved using a novel technique that correlates individual photon hits of events to the known position of the Sun. In an energy window between 0.54 to 0.74 MeV, selected using the dominant scintillation light, we have measured 10 887-2103+2386(stat)±947(syst) (68% confidence interval) solar neutrinos out of 19 904 total events. This corresponds to a Be7 neutrino interaction rate of 51.6-12.5+13.9 counts/(day·100 ton), which is in agreement with the standard solar model predictions and the previous spectroscopic results of Borexino. The no-neutrino hypothesis can be excluded with >5σ confidence level. For the first time, we have demonstrated the possibility of utilizing the directional Cherenkov information for sub-MeV solar neutrinos, in a large-scale, high light yield liquid scintillator detector. This measurement provides an experimental proof of principle for future hybrid event reconstruction using both Cherenkov and scintillation signatures simultaneously.