Background and Objectives: Neurological manifestations of Coronavirus Disease 2019 (COVID-19) such as olfactory and gustatory disturbance have been reported among convalescent COVID-19 patients. However, scientific data on the prevalence of smell and taste disturbance are lacking. Therefore, we present findings on the degree of smell and taste disturbances among the Armenian population. Methods: Study participants were randomly recruited and then categorized into two groups based on their course of the disease. A cross-sectional study was performed to assess participants’ sensitivity to smell triggered by the olfactory and the trigeminal nerves; their ability to differentiate between various odors; and to evaluate their gustatory perception. Results: The smell test revealed that the degree of olfactory nerve disturbance was different by 30.7% in those participants of the early group as compared to those of the late group, and the degree of trigeminal nerve disturbance was different by 71.3% in the early group as compared to the late group. A variation of the differentiating ability among the participants of the early and late groups was detected. Gustatory disturbances for all flavors were also found to be different in both the groups. A moderate positive correlation (0.51) was found between the overall sensitivity of smell and the ability to differentiate between various odors as cumulatively stimulated by both the olfactory and trigeminal nerves. Also, a moderate positive correlation (0.33) was found between headache and smell sensitivity through the olfactory nerve and a high negative correlation (−0.71) was found between headache and smell sensitivity through the trigeminal nerve. Conclusion: Pathological changes in the olfactory and trigeminal perceptive abilities caused disturbances in smell sensation, with the trigeminal nerve being more affected. The capacity to differentiate fragrances did not improve with time and the disturbance severity of bitter taste perception was higher among the study participants.