A major challenge in extracting information from current and upcoming surveys of cosmological Large-Scale Structure (LSS) is the limited availability of computationally expensive high-fidelity simulations. We introduce calibrated Neural Quantile Estimation (NQE), a new Simulation-Based Inference (SBI) method that leverages a large number of approximate simulations for training and a small number of high-fidelity simulations for calibration. This approach guarantees an unbiased posterior regardless of approximate simulation accuracy, while achieving near-optimal constraining power when the approximate simulations are reasonably accurate. As a proof of concept, we demonstrate that cosmological parameters can be inferred at field level from projected 2-dim dark matter density maps up to $k_{\rm max}\sim1.5\,h$/Mpc at $z=0$ by training on $\sim10^4$ Particle-Mesh (PM) simulations with transfer function correction and calibrating with $\sim10^2$ Particle-Particle (PP) simulations. The calibrated posteriors closely match those obtained by directly training on $\sim10^4$ expensive PP simulations, but at a fraction of the computational cost. Our method offers a practical and scalable framework for SBI of cosmological LSS, enabling precise inference across vast volumes and down to small scales.