The coronavirus disease outbreak in 2019 (COVID-19) has now achieved the level of a global pandemic and affected more than 100 million people on all five continents and caused over 2 million deaths. Russia is, needless to say, among the countries affected by SARS-CoV-2, and its health authorities have mobilized significant efforts and resources to fight the disease. The paper presents the result of a functional analysis of 155 patients in the Moscow Region who were examined at the Central Clinical Hospital of the Russian Academy of Sciences during the first wave of the pandemic (February–July, 2020). The inclusion criteria were a positive PCR test and typical, computed tomographic findings of viral pneumonia in the form of ground-glass opacities. A clinical correlation analysis was performed in four groups of patients: (1) those who were not on mechanical ventilation, (2) those who were on mechanical ventilation, and (3) those who subsequently recovered or (4) died. The correlation analysis also considered confounding comorbidities (diabetes, metabolic syndrome, hypertension, etc.). The immunological status of the patients was examined (levels of immunoglobulins of the M, A, G classes and their subclasses, as well as the total immunoglobulin level) using an original SARS-CoV-2 antibody ELISA kit. The ELISA kit was developed using linear S-protein RBD-SD1 and NTD fragments, as well as the N-protein, as antigens. These antigens were produced in the prokaryotic E. coli system. Recombinant RBD produced in the eukaryotic CHO system (RBD CHO) was used as an antigen representing conformational RBD epitopes. The immunoglobulin A level was found to be the earliest serological criterion for the development of a SARS-CoV-2 infection and it yielded the best sensitivity and diagnostic significance of ELISA compared to that of class M immunoglobulin. We demonstrated that the seroconversion rate of “early” N-protein-specific IgM and IgA antibodies is comparable to that of antibodies specific to RBD conformational epitopes. At the same time, seroconversion of SARS-CoV-2 N-protein-specific class G immunoglobulins was significantly faster compared to that of other specific antibodies. Our findings suggest that the strong immunogenicity of the RBD fragment is for the most part associated with its conformational epitopes, while the linear RBD and NTD epitopes have the least immunogenicity. An analysis of the occurrence rate of SARS-CoV-2-specific immunoglobulins of different classes revealed that RBD- and N-specific antibodies should be evaluated in parallel to improve the sensitivity of ELISA. An analysis of the immunoglobulin subclass distribution in sera of seropositive patients revealed uniform induction of N-protein-specific IgG subclasses G1–G4 and IgA subclasses A1–A2 in groups of patients with varying severity of COVID-19. In the case of the S-protein, G1, G3, and A1 were the main subclasses of antibodies involved in the immune response.