The peopling of Sahul (the combined continent of Australia and New Guinea) represents the earliest continental migration and settlement event of solely anatomically modern humans, but its patterns and ecological drivers remain largely conceptual in the current literature. We present an advanced stochastic-ecological model to test the relative support for scenarios describing where and when the first humans entered Sahul, and their most probable routes of early settlement. The model supports a dominant entry via the northwest Sahul Shelf first, potentially followed by a second entry through New Guinea, with initial entry most consistent with 50,000 or 75,000 years ago based on comparison with bias-corrected archaeological map layers. The model's emergent properties predict that peopling of the entire continent occurred rapidly across all ecological environments within 156-208 human generations (4368-5599 years) and at a plausible rate of 0.71-0.92 km year-1. More broadly, our methods and approaches can readily inform other global migration debates, with results supporting an exit of anatomically modern humans from Africa 63,000-90,000 years ago, and the peopling of Eurasia in as little as 12,000-15,000 years via inland routes.