The MicroBooNE liquid argon time projection chamber (LArTPC) maintains a high level of liquid argon purity through the use of a filtration system that removes electronegative contaminants in continuously-circulated liquid, recondensed boil off, and externally supplied argon gas. We use the MicroBooNE LArTPC to reconstruct MeV-scale radiological decays. Using this technique we measure the liquid argon filtration system's efficacy at removing radon. This is studied by placing a 500 kBq $^{222}$Rn source upstream of the filters and searching for a time-dependent increase in the number of radiological decays in the LArTPC. The liquid argon filtration system was able to remove greater than 99.999% of the radon injected into the system. Furthermore, a radiological survey of the filters found that the copper-based filter material was the primary medium that removed the $^{222}$Rn. This is the first observation of radon mitigation in liquid argon with a large-scale copper-based filter and could offer a radon mitigation solution for future large LArTPCs.