Knowing human papillomavirus (HPV) status has important consequences for treatment selection in oropharyngeal cancer. The gold standard is to perform a biopsy. The objective of this paper is to develop a new computed tomography (CT) radiomics-based non-invasive solution to HPV status determination and investigate if and how it can be a viable and accurate complementary technique. Two hundred thirty-eight patients’ CT scans were normalized and resampled. One thousand one hundred forty-two radiomics features were obtained from the segmented CT scans. The number of radiomic attributes was decreased by applying correlation coefficient analysis, backward elimination, and random forest feature importance analysis. Random over-sampling (ROSE) resampling algorithm was performed on the training set for data balancing, and as a result, 161 samples were obtained for each of the HPV classes of the training set. A random forest (RF) classification algorithm was used as a prediction model using five-fold cross-validation (CV). Model effectiveness was evaluated on the unused 20% of the imbalanced data. The applicability of the model was investigated based on previous research and error rates reported for biopsy procedures. The HPV status was determined with an accuracy of 91% (95% CI 83–99) and an area under the curve (AUC) of 0.77 (95% CI 65–89) on the test data. The error rates were comparable to those encountered in biopsy. As a conclusion, radiomics has the potential to predict HPV status with accuracy levels that are comparable to biopsy. Future work is needed to improve standardization, interpretability, robustness, and reproducibility before clinical translation.