Well-dated moraine chronologies indicate that New Zealand glaciers retreated between the onset and the peak of the global Last Glacial Maximum (30–18 ka). The cause of New Zealand net glacier decline during what is thought to have been a ubiquitously cold climate is not known. Here we use a glacier modeling approach to explain the remarkable demise of a former glacier in the Cobb River valley (South Island), which retreated by >50% prior to 20 ka, and may have melted completely by 18 ka—prior to the onset of rapid deglaciation evident across the southern mid-latitudes during Heinrich Stadial 1. Modeling experiments show that this glacier had an amplified response to modest climate forcing, requiring only a minor atmospheric warming (+0.5 °C) or precipitation reduction (>15%) to force retreat. Our results, when considered alongside pollen and speleothem evidence from the region, indicate that minor interstadial warming and/or drying of the atmosphere in New Zealand drove regional glacier retreat prior to 20 ka. Our results reinforce that mountain glaciers are sensitive indicators of climate change—but highlight that some are more sensitive than others. This high-sensitivity glacier record provides quantitative constraint of low-magnitude climate forcing that immediately preceded the abrupt Southern Hemisphere transition out of the Last Glacial Maximum.