Robust entanglement at room temperature is a necessary requirement for practical applications in quantum technology. We demonstrate the creation of bipartite- and tripartite-entangled quantum states in a small quantum register consisting of individual 13C nuclei in a diamond lattice. Individual nuclear spins are controlled via their hyperfine coupling to a single electron at a nitrogen-vacancy defect center. Quantum correlations are of high quality and persist on a millisecond time scale even at room temperature, which is adequate for sophisticated quantum operations.