One of the primary reasons that superconductivity in underdoped cuprates is enigmatic is that it emerges from an incoherent Fermi-arc state, so the applicability of the Bardeen-Cooper-Schrieffer (BCS) theory is questionable. Here we approach this problem by investigating unconventional d-wave superconductivity in a recently proposed solvable model for strongly correlated Fermi arcs. We show analytically that the exact incorporation of Fermi arcs fundamentally modifies the BCS equations, which enables us to isolate a many-body effect that suppresses the superconducting transition temperature T_c beyond the simple reduction expected from a shrinking Fermi surface. The theory unambiguously produces: (i) a T_c tracing out a dome as a function of hole doping, (ii) a new low-energy mode upon entering superconductivity, (iii) a suppressed superfluid stiffness in the underdoped regime, and (iv) a gap-to-T_c ratio far exceeding the BCS limit, all consistent with experimental observations in cuprate superconductors. These findings provide an analytic benchmark for understanding how superconductivity emerges from a correlated Fermi-arc state in high-T_c superconductors.