Summary We present a methodical procedure for topology optimization under uncertainty with multiresolution finite element (FE) models. We use our framework in a bifidelity setting where a coarse and a fine mesh corresponding to low‐ and high‐resolution models are available. The inexpensive low‐resolution model is used to explore the parameter space and approximate the parameterized high‐resolution model and its sensitivity, where parameters are considered in both structural load and stiffness. We provide error bounds for bifidelity FE approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems, where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability, reliability‐based topology optimization, and three‐dimensional topology optimization while generating almost identical designs to those obtained with a single‐resolution mesh. We also compute the parametric von Mises stress for the generated designs via our bifidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm, which extends the well‐known 88‐line topology optimization code in MATLAB, is provided.