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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 degrees 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.