Liquid scintillator detectors play a central role in the detection of neutrinos from various sources. In particular, it is the only technique used so far for the precision spectroscopy of sub-MeV solar neutrinos, as demonstrated by the Borexino experiment at the Gran Sasso National Laboratory in Italy. The benefit of a high light yield, and thus a low energy threshold and a good energy resolution, comes at the cost of the directional information featured by water Cherenkov detectors, measuring $^8$B solar neutrinos above a few MeV. In this paper we provide the first directionality measurement of sub-MeV solar neutrinos which exploits the correlation between the first few detected photons in each event and the known position of the Sun for each event. This is also the first signature of directionality in neutrinos elastically scattering off electrons in a liquid scintillator target. This measurement exploits the sub-dominant, fast Cherenkov light emission that precedes the dominant yet slower scintillation light signal. Through this measurement, we have also been able to extract the rate of $^{7}$Be solar neutrinos in Borexino. The demonstration of directional sensitivity in a traditional liquid scintillator target paves the way for the possible exploitation of the Cherenkov light signal in future kton-scale experiments using liquid scintillator targets. Directionality is important for background suppression as well as the disentanglement of signals from various sources.