Our paper investigates the bioeconomic sustainability and resilience of savanna social-ecological systems (SES). A stylized dynamics of an exploited grass-tree systems is thus considered accounting both for the competition between trees and grass along with logging and grass harvesting activities. Regarding sustainability, we rely on bioeconomic viability goals including consumption security for grass, profitability of logging, and coexistence of tree-grass states. A first analytical result relates to the elicitation of sufficient sustainability conditions through the non-emptiness of the so-called viability kernel. Such sufficient conditions rely on coupled MSY (maximum sustainable yield)-MEY (maximum economic yield) reference states-controls. A larger viable set including these MSY-MEY equilibria is also identified. The resilience of such viability states-controls for savanna SES facing shocks such as fire is then put forward from both recovery through stability analysis and resistance viewpoints. These analytical results provide managers with different upper and lower bioeconomic thresholds on the savanna SESs to promote their sustainability and resilience. Simulations inspired from savanna systems in Cameroon exemplify the analytical findings.