We study the scaling properties of the entanglement entropy (EE) near quantum critical points in interacting random antiferromagnetic (AF) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between Haldane and Random Singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge $c_{\rm eff} = 1.17(4)$ at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $ν=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on Imry-Ma argument. We provide a possible explanation.
