The nearby radio galaxy Centaurus A belongs to a class of active galaxies that are luminous at radio wavelengths. Most show collimated relativistic outflows known as jets, which extend over hundreds of thousands of parsecs for the most powerful sources. Accretion of matter onto the central supermassive black hole is believed to fuel these jets and power their emission1. Synchrotron radiation from relativistic electrons causes the radio emission, and it has been suggested that the X-ray emission from Centaurus A also originates in electron synchrotron processes2-4. Another possible explanation is inverse Compton scattering with cosmic microwave background (CMB) soft photons5-7. Synchrotron radiation needs ultrarelativistic electrons (about 50 teraelectronvolts) and, given their short cooling times, requires some continuous re-acceleration mechanism8. Inverse Compton scattering, on the other hand, does not require very energetic electrons, but the jets must stay highly relativistic on large scales (exceeding 1 megaparsec). Some recent evidence disfavours inverse Compton-CMB models9-12, although other work seems to be compatible with them13,14. In principle, the detection of extended γ-ray emission, which directly probes the presence of ultrarelativistic electrons, could distinguish between these options. At gigaelectronvolt energies there is also an unusual spectral hardening15,16 in Centaurus A that has not yet been explained. Here we report observations of Centaurus A at teraelectronvolt energies that resolve its large-scale jet. We interpret the data as evidence for the acceleration of ultrarelativistic electrons in the jet, and favour the synchrotron explanation for the X-rays. Given that this jet is not exceptional in terms of power, length or speed, it is possible that ultrarelativistic electrons are commonplace in the large-scale jets of radio-loud active galaxies.