Super-massive black holes residing at the centres of galaxies can launch powerful radio-emitting plasma jets which reach scales of hundreds of thousands of light-years, well beyond their host galaxies.1 The advent of Chandra, the only X-ray observatory capable of sub-arcsecond-scale imaging, has lead to the surprising discovery of strong X-ray emission from radio jets on these scales.2-4 The origin of this X-ray emission, which appears as a second spectral component from that of the radio emission, has been debated for over two decades.5-9 The most commonly assumed mechanism is inverse Compton upscattering of the Cosmic Microwave Background (IC-CMB) by very low-energy electrons in a still highly relativistic jet.10,11 Under this mechanism no variability in the X-ray emission is expected. Here we report the detection of X-ray variability in the large-scale jet population, using a novel statistical analysis of 53 jets with multiple Chandra observations. Individually 13/53 jets have at least one feature which is variable at the p<0.05 level. Taken as a population, we find that the distribution of p-values from a Poisson model is strongly inconsistent with steady emission, with a global p-value of 1.96×10−4 under a Kolmogorov-Smirnov test against the expected Uniform (0,1) distribution. The inconsistency significantly increases when we exclude core-dominated quasars at high redshift. These results strongly imply that the dominant mechanism of X-ray production in kpc-scale jets is synchrotron emission by a second population of electrons reaching multi-TeV energies. X-ray variability on the time-scale of months to a few years implies extremely small emitting volumes much smaller than the cross-section of the jet.