We describe the development of a noise-temperature testing capability for phased-array antennas operating in receive mode from 0.7 GHz to 1.8 GHz. Sampled voltages from each array port were recorded digitally as the zenith-pointing array under test was presented with three scenes: (1) a large microwave absorber at ambient temperature, (2) the unobstructed radio sky, and (3) broadband noise transmitted from a reference antenna centred over and pointed at the array under test. The recorded voltages were processed in software to calculate the beam equivalent noise temperature for a maximum signal-to-noise ratio beam steered at the zenith. We introduced the reference-antenna measurement to make noise measurements with reproducible, well-defined beams directed at the zenith and thereby at the centre of the absorber target. We applied a detailed model of cosmic and atmospheric contributions to the radio sky emission that we used as a noise-temperature reference. We also present a comprehensive analysis of measurement uncertainty including random and systematic effects. The key systematic effect was due to uncertainty in the beamformed antenna pattern and how efficiently it illuminates the absorber load. We achieved a combined uncertainty as low as 4 K for a 40 K measurement of beam equivalent noise temperature. The measurement and analysis techniques described in this paper were pursued to support noise-performance verification of prototype phased-array feeds for the Australian Square Kilometre Array Pathfinder telescope.