Super-massive black holes in active galaxies can accelerate particles to relativistic energies, producing jets with associated gamma-ray emission. Galactic 'microquasars', which are binary systems consisting of a neutron star or stellar-mass black hole accreting gas from a companion star, also produce relativistic jets, generally together with radio flares. Apart from an isolated event detected in Cygnus X-1, there has hitherto been no systematic evidence for the acceleration of particles to gigaelectronvolt or higher energies in a microquasar, with the consequence that we are as yet unsure about the mechanism of jet energization. Here we report four gamma-ray flares with energies above 100 MeV from the microquasar Cygnus X-3 (an exceptional X-ray binary that sporadically produces radio jets). There is a clear pattern of temporal correlations between the gamma-ray flares and transitional spectral states of the radio-frequency and X-ray emission. Particle acceleration occurred a few days before radio-jet ejections for two of the four flares, meaning that the process of jet formation implies the production of very energetic particles. In Cygnus X-3, particle energies during the flares can be thousands of times higher than during quiescent states.