HPC systems face security and compliance challenges, particularly in preventing waste and misuse of computational resources by unauthorized or malicious software that deviates from allocation purpose. Existing methods to classify applications based on job names or resource usage are often unreliable or fail to capture applications that have different behavior due to different inputs or system noise. This research proposes an approach that uses similarity-preserving fuzzy hashes to classify HPC application executables. By comparing the similarity of SSDeep fuzzy hashes, a Random Forest Classifier can accurately label applications executing on HPC systems including unknown samples. We evaluate the Fuzzy Hash Classifier on a dataset of 92 application classes and 5333 distinct application samples. The proposed method achieved a macro f1-score of 90% (micro f1-score: 89%, weighted f1-score: 90%). Our approach addresses the critical need for more effective application classification in HPC environments, minimizing resource waste, and enhancing security and compliance.