Most general-purpose classification methods, such as support-vector machine (SVM) and random forest (RF), fail to account for an unusual characteristic of astronomical data: known measurement error uncertainties. In astronomical data, this information is often given in the data but discarded because popular machine learning classifiers cannot incorporate it. We propose a simulation-based approach that incorporates heteroscedastic measurement error into any existing classification method to better quantify uncertainty in classification. The proposed method first simulates perturbed realizations of the data from a Bayesian posterior predictive distribution of a Gaussian measurement error model. Then, a chosen classifier is fit to each simulation. The variation across the simulations naturally reflects the uncertainty propagated from the measurement errors in both labeled and unlabeled data sets. We demonstrate the use of this approach via two numerical studies. The first is a thorough simulation study applying the proposed procedure to SVM and RF, which are well-known hard and soft classifiers, respectively. The second study is a realistic classification problem of identifying high-redshift quasar candidates from photometric data. The proposed approach reveals that out of 11,847 high-redshift quasar candidates identified by a random forest without incorporating measurement error, 3,146 are potential misclassifications. Additionally, out of ~1.85 million objects not identified as high-redshift quasars without measurement error, 936 can be considered candidates when measurement error is taken into account.