Energetic electrons and ions that are trapped in Earth's magnetosphere can suddenly be accelerated towards the planet. Some dynamic features of Earth's aurora (the northern and southern lights) are created by the fraction of these injected particles that travels along magnetic field lines and hits the upper atmosphere. Jupiter's aurora appears similar to Earth's in some respects; both appear as large ovals circling the poles and both show transient events. But the magnetospheres of Jupiter and Earth are so different---particularly in the way they are powered---that it is not known whether the magnetospheric drivers of Earth's aurora also cause them on Jupiter. Here we show a direct relationship between Earth-like injections of electrons in Jupiter's magnetosphere and a transient auroral feature in Jupiter's polar region. This relationship is remarkably similar to what happens at Earth, and therefore suggests that despite the large differences between planetary magnetospheres, some processes that generate aurorae are the same throughout the Solar System.