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Computational models are utilized in many scientific domains to simulate complex systems. Sensitivity analysis is an important practice to aid our understanding of the mechanics of these models and the processes they describe, but performing a sufficient number of model evaluations to obtain accurate sensitivity estimates can often be prohibitively expensive. In order to reduce the computational burden, a common solution is to use a surrogate model that approximates the original model reasonably well but at a fraction of the cost. However, in exchange for the computational benefits of surrogate-based sensitivity analysis, this approach comes with the price of a loss in accuracy arising from the difference between the surrogate and the original model. To address this issue, we adapt the floodgate method of Zhang and Janson (2020) to provide valid surrogate-based confidence intervals rather than a point estimate, allowing for the benefit of the computational speed-up of using a surrogate that is especially pronounced for high-dimensional models, while still retaining rigorous and accurate bounds on the global sensitivity with respect to the original (non-surrogate) model. Our confidence interval is asymptotically valid with almost no conditions on the computational model or the surrogate. Additionally, the accuracy (width) of our confidence interval shrinks as the surrogate's accuracy increases, so when an accurate surrogate is used, the confidence interval we report will correspondingly be quite narrow, instilling appropriately high confidence in its estimate. We demonstrate the properties of our method through numerical simulations on the small Hymod hydrological model, and also apply it to the more complex CBM-Z meteorological model with a recent neural-network-based surrogate.