White dwarfs are often found in binary systems with orbital periods ranging from tens of minutes to hours in which they can accrete gas from their companion stars. In about 15 per cent of these binaries, the magnetic field of the white dwarf is strong enough (at 10 gauss or more) to channel the accreted matter along field lines onto the magnetic poles. The remaining systems are referred to as 'non-magnetic', because until now there has been no evidence that they have a magnetic field that is strong enough to affect the accretion dynamics. Here we report an analysis of archival optical observations of the 'non-magnetic' accreting white dwarf in the binary system MV Lyrae, whose light curve displays quasi-periodic bursts of about 30 minutes duration roughly every 2 hours. The timescale and amplitude of these bursts indicate the presence of an unstable, magnetically regulated accretion mode, which in turn implies the existence of magnetically gated accretion, in which disk material builds up around the magnetospheric boundary (at the co-rotation radius) and then accretes onto the white dwarf, producing bursts powered by the release of gravitational potential energy. We infer a surface magnetic field strength for the white dwarf in MV Lyrae of between 2 × 10 gauss and 1 × 10 gauss, too low to be detectable by other current methods. Our discovery provides a new way of studying the strength and evolution of magnetic fields in accreting white dwarfs and extends the connections between accretion onto white dwarfs, young stellar objects and neutron stars, for which similar magnetically gated accretion cycles have been identified.