Arrays of covalently bound organic molecules possess potential for light-harvesting and energy transfer applications due to the strong coherent dipole-dipole coupling between the transition dipole moments of the molecules involved. Here, we show that such molecular systems, based on perylene molecules, can be considered as arrays of qubits that are amenable for laser-driven quantum coherent control. The perylene monomers exhibit dephasing times four orders of magnitude longer than a typical gating time, thus allowing for the execution of a large number of gate operations on the subpicosecond timescale. Specifically, we demonstrate quantum logic gates and entanglement in bipartite (dimer) and tripartite (trimer) systems of perylene-based arrays. In dimers, naturally entangled states with a tailored degree of entanglement can be produced. The nonlocality of the molecular trimer entanglement is demonstrated by testing Mermin's (Bell-like) inequality violation.