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Atrioventricular septal defect (AVSD) is a clinically significant subtype of congenital heart disease (CHD) that severely influences the health of babies during birth and is associated with Down syndrome (DS). Thus, exploring the differences in functional genes in DS samples with and without AVSD is a critical way to investigate the complex association between AVSD and DS. In this study, we present a computational method to distinguish DS patients with AVSD from those without AVSD using the newly proposed self-normalizing neural network (SNN). First, each patient was encoded by using the copy number of probes on chromosome 21. The encoded features were ranked by the reliable Monte Carlo feature selection (MCFS) method to obtain a ranked feature list. Based on this feature list, we used a two-stage incremental feature selection to construct two series of feature subsets and applied SNNs to build classifiers to identify optimal features. Results show that 2737 optimal features were obtained, and the corresponding optimal SNN classifier constructed on optimal features yielded a Matthew’s correlation coefficient (MCC) value of 0.748. For comparison, random forest was also used to build classifiers and uncover optimal features. This method received an optimal MCC value of 0.582 when top 132 features were utilized. Finally, we analyzed some key features derived from the optimal features in SNNs found in literature support to further reveal their essential roles.