Despite its current relatively low global share of cases and deaths in Africa compared to other regions, coronavirus disease 2019 (COVID-19) has the potential to trigger other larger crises in the region. This is due to the vulnerability of health and economic systems, coupled with the high burden of human immunodeficiency virus (HIV), tuberculosis (TB), and malaria. Here we examine the potential implications of COVID-19 on the control of these major epidemic diseases in Africa. We use current evidence on disease burden of HIV, TB, and malaria, and epidemic dynamics of COVID-19 in Africa, retrieved from the literature. Our analysis shows that the current measures to control COVID-19 neglect important and complex context-specific epidemiological, social, and economic realities in Africa. There is a similarity of clinical features of TB and malaria, with those used to track COVID-19 cases. This coupled with institutional mistrust and misinformation might result in many patients with clinical features similar to those of COVID-19 being hesitant to voluntarily seek care in a formal health facility. Furthermore, most people in productive age in Africa work in the informal sector, and most of those in the formal sector are underemployed. With the current measures to control COVID-19, these populations might face unprecedented difficulties to access essential services, mainly due to reduced ability of patients to support direct and indirect medical costs, and unavailability of transportation means to reach health facilities. Therefore, if not accompanied with appropriate economic and epidemiological considerations, we anticipate that these measures might result in unprecedented difficulties among vulnerable segments of society to access essential services, including antiretroviral and prophylactic drugs among people living with HIV and Acquired Immune Deficiency Syndrome, anti-tuberculosis drugs, and curative and preventive treatments for malaria among pregnant women and children. This might increase the propensity of patients taking substandard doses and/or medicines, which has the potential to compromise drug efficacy, and worsen health inequalities in the region. COVID-19 responses at country level should include measures to protect vulnerable and under-served segments of society.