Among practitioners, the importance of inputs to a model output is commonly measured via the computation of Sobol' sensitivity indices. Various estimation strategies exist in the literature, most of them requiring a very high number of model evaluations. Designing methods that compete favorably both in terms of computational cost and accuracy is therefore an issue of crucial importance. In this paper, an efficient replication-based strategy is proposed to estimate the full set of first- and second-order Sobol' indices. It relies on a Sobol' pick-freeze estimation scheme and requires only two replicated designs based on randomized orthogonal arrays of strength two. The precision of this procedure is assessed with bootstrap confidence intervals, presented for the first time in the replication framework. Our developments are compared to known approaches and validated on numerical test cases. A way to estimate the full set of first-, second-order but also total-effect Sobol' indices at a very competitive cost is also described, as a combination of our procedure and the one introduced by Saltelli.