The Nernst effect in superconductors is typically linked to fluctuating Cooper pairs above T_c or vortex motion below T_c. We show instead that Berry curvature of Bogoliubov quasiparticles can generate an intrinsic Nernst response in a clean, vortex-free superconducting state. Focusing on two-dimensional systems with Ising spin-orbit coupling, relevant to transition-metal dichalcogenides, we identify two regimes: an intervalley s-wave paired state where a weak magnetic field activates the effect, and an intravalley chiral p-wave paired state that exhibits a spontaneous charge or spin Nernst response without a field. We propose an experimental setup that circumvents screening and provide estimates of the signal magnitude. Our results establish the Nernst effect as a direct probe of Berry curvature and pairing symmetry in two-dimensional spin-orbit-coupled superconductors.