Over the last few decades, many economies in sub-Saharan Africa have experienced much faster economic growth than other parts of the world. However, many of these economies have not experienced significant poverty reduction. Several factors such as the quality of governance may limit the expected effects of economic growth on poverty. This paper examines the triangular relationship between extreme poverty, governance quality, and economic growth for the sub-Saharan African countries over the period 2010–2019. Compared to the work carried out until now, the novelty of this research lies in using the Panel Threshold Regression (PTR) and Panel Smooth Transition Regression (PSTR) models to determine the optimal level of governance index, which once attained, will make extreme poverty decrease with economic growth and governance quality. We found that the nexus between these three variables is nonlinear. Besides, results show that there exists a statistically negative relationship between governance and extreme poverty above the threshold level of 0.314 for the Global Governance Index (GGI) and 66.9 for the Ibrahim Index of African Governance (IIAG), above which governance quality decreases extreme poverty. The results showed that the economic growth would begin to reduce extreme poverty once governance reaches a threshold level of 0.367 for GGI and 63.2 for IIAG. Better performance of governance also appears to improve economic growth and reduces poverty.