We consider the problem of learning an ε-optimal policy in controlled dynamical systems with low-rank latent structure. For this problem, we present LoRa-PI (Low-Rank Policy Iteration), a model-free learning algorithm alternating between policy improvement and policy evaluation steps. In the latter, the algorithm estimates the low-rank matrix corresponding to the (state, action) value function of the current policy using the following two-phase procedure. The entries of the matrix are first sampled uniformly at random to estimate, via a spectral method, the leverage scores of its rows and columns. These scores are then used to extract a few important rows and columns whose entries are further sampled. The algorithm exploits these new samples to complete the matrix estimation using a CUR-like method. For this leveraged matrix estimation procedure, we establish entry-wise guarantees that remarkably, do not depend on the coherence of the matrix but only on its spikiness. These guarantees imply that LoRa-PI learns an ε-optimal policy using Õ(S+A/poly(1 - γ)ε2) samples where S (resp. A) denotes the number of states (resp. actions) and γ the discount factor. Our algorithm achieves this order-optimal (in S, A and ε) sample complexity under milder conditions than those assumed in previously proposed approaches.