We have used the Australia Telescope Compact Array (ATCA) at 95GHz to carry out continuum observations of 130 extragalactic radio sources selected from the Australia Telescope 20GHz (AT20G) survey. We use a triple-correlation method to measure simultaneous 20- and 95-GHz flux densities for these objects, and over 90 per cent of our target sources are detected at 95GHz. We demonstrate that the ATCA can robustly measure 95-GHz flux densities with an accuracy of ~10 per cent in a few minutes for sources stronger than about 50mJy. We measure the distribution of radio spectral indices in a flux-limited sample of extragalactic sources, and show that the median 20-95GHz spectral index does not vary significantly with flux density for S_20_>150mJy . This finding allows us to estimate the extragalactic radio source counts at 95GHz by combining our observed 20-95GHz spectral-index distribution with the accurate 20-GHz source counts measured in the AT20G survey.