Neuroimaging data from multiple batches (i.e. acquisition sites, scanner manufacturer, datasets, etc.) are increasingly necessary to gain new insights into the human brain. However, multi-batch data, as well as extracted radiomic features, exhibit pronounced technical artifacts across batches. These batch effects introduce confounding into the data and can obscure biological effects of interest, decreasing the generalizability and reproducibility of findings. This is especially true when multi-batch data is used alongside complex downstream analysis models, such as machine learning methods. Image harmonization methods seeking to remove these batch effects are important for mitigating these issues; however, significant multivariate batch effects remain in the data following harmonization by current state-of-the-art statistical and deep learning methods. We present DeepCombat, a deep learning harmonization method based on a conditional variational autoencoder architecture and the ComBat harmonization model. DeepCombat learns and removes subject-level batch effects by accounting for the multivariate relationships between features. Additionally, DeepComBat relaxes a number of strong assumptions commonly made by previous deep learning harmonization methods and is empirically robust across a wide range of hyperparameter choices. We apply this method to neuroimaging data from a large cognitive-aging cohort and find that DeepCombat outperforms existing methods, as assessed by a battery of machine learning methods, in removing scanner effects from cortical thickness measurements while preserving biological heterogeneity. Additionally, DeepComBat provides a new perspective for statistically-motivated deep learning harmonization methods.