Although the upstream translation of waterfalls is often thought to occur by undercutting of resistant strata, collapse, and headwall retreat (e.g., Niagara Falls), many propagating waterfalls maintain a vertical face in the absence of undercutting. To explain this observation, we propose that bedrock-fracture geometry exerts a fundamental control on knickpoint morphology and evolution such that vertical waterfalls can persist during retreat due to toppling in bedrock with near horizontal and vertical sets of joints (e.g., columnar basalt). At a waterfall, rock columns are affected by shear and drag from the overflowing water, buoyancy from the plunge pool at the foot of the waterfall, and gravity. We used a torque balance to determine the stability of a rock column and any individual blocks that comprise the column. Results indicate that rotational failure should occur about the base of a headwall (and therefore preserve its form during upstream propagation) where columns are tilted in the downstream direction or slightly tilted in the upstream direction, depending on the plunge-pool depth. Flume experiments were performed to test the model, and the model provides a good prediction of the flow necessary to induce toppling and the morphology of the head-wall. Waterfall-induced toppling explains the morphology of canyon headwalls in the volcanic terrain of the northwestern United States, where catastrophic paleofloods (e.g., Bonneville Flood) have carved steep amphitheater-headed canyons in columnar basalt. This model may also explain similar land-forms elsewhere on Earth and Mars, and it can be used to predict the minimum flow discharge needed to create these features.