Moringa oleifera (M. oleifera) is a plant widely used for its beneficial properties both in medical and non-medical fields. Because they produce bioactive metabolites, plants are a major resource for drug discovery. In this study, two different cultivars of leaves of M. oleifera (Salento and Barletta) were obtained by maceration or microwave-assisted extraction (MAE). We demonstrated that extracts obtained by MAE exhibited a lower cytotoxic profile compared to those obtained by maceration at concentrations ranged from 25 to 400 µg/mL, on both Vero CCL-81 and Vero/SLAM cells. We examined their antiviral properties against two viruses, i.e., the human coronavirus 229E (HCoV-229E) and measles virus (MeV), which are both responsible for respiratory infections. The extracts were able to inhibit the infection of both viruses and strongly prevented their attack and entry into the cells in a range of concentrations from 50 to 12 µg/mL. Particularly active was the variety of Salento that registered a 50% inhibitory concentration (IC50) at 21 µg/mL for HCoV-229E and at 6 µg/mL for MeV. We identified the presence of several compounds through high performance liquid chromatography (HPLC); in particular, chlorogenic and neochlorogenic acids, quercetin 3-O-β-d-glucopyranoside (QGP), and glucomoringin (GM) were mainly observed. In the end, M. oleifera can be considered a promising candidate for combating viral infections with a very strong action in the early stages of viral life cycle, probably by destructuring the viral particles blocking the virus–cell fusion.