The last deglaciation (21 to 7 thousand years ago) was punctuated by several abrupt meltwater pulses, which sometimes caused noticeable climate change. Around 14 thousand years ago, meltwater pulse 1A (MWP-1A), the largest of these events, produced a sea level rise of 14-18 metres over 350 years. Although this enormous surge of water certainly originated from retreating ice sheets, there is no consensus on the geographical source or underlying physical mechanisms governing the rapid sea level rise. Here we present an ice-sheet modelling simulation in which the separation of the Laurentide and Cordilleran ice sheets in North America produces a meltwater pulse corresponding to MWP-1A. Another meltwater pulse is produced when the Labrador and Baffin ice domes around Hudson Bay separate, which could be associated with the `8,200-year' event, the most pronounced abrupt climate event of the past nine thousand years. For both modelled pulses, the saddle between the two ice domes becomes subject to surface melting because of a general surface lowering caused by climate warming. The melting then rapidly accelerates as the saddle between the two domes gets lower, producing nine metres of sea level rise over 500 years. This mechanism of an ice `saddle collapse' probably explains MWP-1A and the 8,200-year event and sheds light on the consequences of these events on climate.