Functional near-infrared spectroscopy-based brain-computer interface (fNIRS-based BCI) has been receiving much attention. However, we are practically constrained to obtain a lot of fNIRS data by inherent hemodynamic delay. For this reason, when employing machine learning techniques, a problem due to the high-dimensional feature vector may be encountered, such as deteriorated classification accuracy. In this study, we employ an elastic net-based feature selection which is one of the embedded methods and demonstrate the utility of which by analyzing the results. Using the fNIRS dataset obtained from 18 participants for classifying brain activation induced by mental arithmetic and idle state, we calculated classification accuracies after performing feature selection while changing the parameter α (weight of lasso vs. ridge regularization). Grand averages of classification accuracy are 80.0 ± 9.4%, 79.3 ± 9.6%, 79.0 ± 9.2%, 79.7 ± 10.1%, 77.6 ± 10.3%, 79.2 ± 8.9%, and 80.0 ± 7.8% for the various values of α = 0.001, 0.005, 0.01, 0.05, 0.1, 0.2, and 0.5, respectively, and are not statistically different from the grand average of classification accuracy estimated with all features (80.1 ± 9.5%). As a result, no difference in classification accuracy is revealed for all considered parameter α values. Especially for α = 0.5, we are able to achieve the statistically same level of classification accuracy with even 16.4% features of the total features. Since elastic net-based feature selection can be easily applied to other cases without complicated initialization and parameter fine-tuning, we can be looking forward to seeing that the elastic-based feature selection can be actively applied to fNIRS data.