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Unexpectedly strong X-ray emission from extragalactic radio jets on kiloparsec scales has been one of the major discoveries of Chandra, the only X-ray observatory capable of sub-arcsecond-scale imaging. 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. The most commonly assumed mechanism is inverse-Compton upscattering of the cosmic microwave background by very low-energy electrons in a still highly relativistic jet. 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. 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. These results strongly imply that the dominant mechanism of X-ray production in kiloparsec-scale jets is synchrotron emission by a second population of electrons reaching multi-teraelectronvolt energies. X-ray variability on the timescale of months to a few years implies extremely small emitting volumes much smaller than the cross-section of the jet.