Many palaeoclimate records from the North Atlantic region show a pattern of rapid climate oscillations, the so-called Dansgaard-Oeschger events, with a quasi-periodicity of approximately 1,470 years for the late glacial period. Various hypotheses have been suggested to explain these rapid temperature shifts, including internal oscillations in the climate system and external forcing, possibly from the Sun. But whereas pronounced solar cycles of approximately 87 and approximately 210 years are well known, a approximately 1,470-year solar cycle has not been detected. Here we show that an intermediate-complexity climate model with glacial climate conditions simulates rapid climate shifts similar to the Dansgaard-Oeschger events with a spacing of 1,470 years when forced by periodic freshwater input into the North Atlantic Ocean in cycles of approximately 87 and approximately 210 years. We attribute the robust 1,470-year response time to the superposition of the two shorter cycles, together with strongly nonlinear dynamics and the long characteristic timescale of the thermohaline circulation. For Holocene conditions, similar events do not occur. We conclude that the glacial 1,470-year climate cycles could have been triggered by solar forcing despite the absence of a 1,470-year solar cycle.