Enveloped viruses such as HIV-1 enter their hosts by first establishing a contact region at the cell surface, which is stabilized by the formation of receptor-ligand complexes. We show that the favorable contact energy stemming from the formation of the receptor complexes in the interaction zone is sufficient to drive the engulfment of the virus by the cell. Using a continuum model, we show that the equilibrium engulfment depth and the force driving the engulfment are functions of the virus size and the complex formation energy. Resistance to engulfment is dominated by the elastic deformation of the cytoskeleton.