The Auger Engineering Radio Array (AERA) is situated in the Argentinian Pampa Amarilla, a location far away from large human settlements. Nevertheless, a strong background of pulsed radio-frequency interference (RFI) exists on site, which not only makes radio self-triggering challenging but also poses a problem for an efficient and pure reconstruction of air-shower measurements. We present how our standard event reconstruction exploits several strategies to identify and suppress pulsed noise, and quantify the efficiency and purity of our algorithms. These strategies can be employed by any experiment taking radio data in the presence of pulsed RFI.