Brain disease can be screened using eye movements. Degenerative brain disorders change eye movement because they affect not only memory and cognition but also the cranial nervous system involved in eye movement. We compared the facial and eye movement patterns of patients with mild Alzheimer’s disease and cognitively normal people to analyze the neurological signs of dementia. After detecting the facial landmarks, the coordinate values for the movements were extracted. We used Spearman’s correlation coefficient to examine associations between horizontal and vertical facial and eye movements. We analyzed the correlation between facial and eye movements without using special eye-tracking equipment or complex conditions in order to measure the behavioral aspect of the natural human gaze. As a result, we found differences between patients with Alzheimer’s disease and cognitively normal people. Patients suffering from Alzheimer’s disease tended to move their face and eyes simultaneously in the vertical direction, whereas the cognitively normal people did not, as confirmed by a Mann–Whitney–Wilcoxon test. Our findings suggest that objective and accurate measurement of facial and eye movements can be used to screen such patients quickly. The use of camera-based testing for the early detection of patients showing signs of neurodegeneration can have a significant impact on the public care of dementia.