Attempts to place Palaeolithic finds within a precise climatic framework are complicated by both uncertainty over the radiocarbon calibration beyond about 21,500 14C years bp and the absence of a master calendar chronology for climate events from reference archives such as Greenland ice cores or speleothems. Here we present an alternative approach, in which 14C dates of interest are mapped directly onto the palaeoclimate record of the Cariaco Basin by means of its 14C series, circumventing calendar age model and correlation uncertainties, and placing dated events in the millennial-scale climate context of the last glacial period. This is applied to different sets of dates from levels with Mousterian artefacts, presumably produced by late Neanderthals, from Gorham's Cave in Gibraltar: first, generally accepted estimates of about 32,000 14C years bp for the uppermost Mousterian levels; second, a possible extended Middle Palaeolithic occupation until about 28,000 14C years bp; and third, more contentious evidence for persistence until about 24,000 14C years bp. This study shows that the three sets translate to different scenarios on the role of climate in Neanderthal extinction. The first two correspond to intervals of general climatic instability between stadials and interstadials that characterized most of the Middle Pleniglacial and are not coeval with Heinrich Events. In contrast, if accepted, the youngest date indicates that late Neanderthals may have persisted up to the onset of a major environmental shift, which included an expansion in global ice volume and an increased latitudinal temperature gradient. More generally, our radiocarbon climatostratigraphic approach can be applied to any 'snapshot' date from discontinuous records in a variety of deposits and can become a powerful tool in evaluating the climatic signature of critical intervals in Late Pleistocene human evolution.