The formation and offshore transport of dense water over a uniformly sloping shelf crosscut by a submarine canyon is examined using a three-dimensional primitive-equation numerical model. A constant negative buoyancy flux is applied in a limited region adjacent to a straight coast to represent brine rejection from ice production in an idealized coastal polynya. A sharp density front forms at the edge of the forcing region, with surface and bottom intensified jets along the front. The flow around the head of the submarine canyon triggers a frontal instability that initially grows only on one side of the canyon. The unstable waves on the other side of the canyon are blocked by a localized barotropic flow that develops near the canyon head. Unstable waves also grow where the forcing region intersects the coast. The frontal waves grow rapidly (with O(1 day) e-folding timescales) and form eddies with horizontal scales of O(15 km) which extract the densest water from the forcing region and carry it offshore, directly across isobaths. In this way the eddies limit the maximum water density that appears in the model despite continued negative buoyancy forcing. Some dense water descends into the canyon, forming a bottom-trapped plume that transports the dense water offshore ahead of the eddies. The plume moves relatively slowly (i.e., small Froude number), with little turbulent entrainment, so the advancement and structure of the plume nose can be described successfully as a simple gravity current with an advective-diffusive heat balance. Eddies may slump into the canyon from the side, altering both the density anomaly and speed of the canyon plume, suggesting that canyon plumes are likely to be highly variable in both space and time.