Supernova (SN) 1987A in the Large Magellanic Cloud (LMC) has proven to be a unique laboratory within which to investigate particle acceleration in young supernova remnants (SNRs). Here we report the first detection of linear polarization of the supernova's synchrotron emission from imaging observations at frequencies spanning from 20 to 50 GHz, carried out with the Australia Telescope Compact Array (ATCA) between 2015 October and 2016 May. The direction of the radio polarization, corrected for Faraday rotation, points to a primarily radial magnetic field across the inner ring, encompassing both the reverse and forward shocks. The magnetic field strength peaks over the high-emissivity eastern sites, where efficient cosmic-ray acceleration likely takes place under quasi-parallel shocks at high Mach numbers. The mean fraction of polarized emission in the brightest sites is 2.7% ±0.2% at 22 GHz and 3.5% ±0.7% at 44 GHz. In the inner remnant, non-radial components of the polarized emission appear to be more prevalent. However, the low significance detection in the central regions limits interpretation.