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. Over 90% of these sources are detected at 95 GHz, and we use a triple-correlation method to measure simultaneous 20 and 95 GHz flux densities. We show that the ATCA can measure 95GHz flux densities to ~10% accuracy in a few minutes for sources above ~50mJy. The median 20-95GHz spectral index does not vary significantly with flux density for extragalactic sources with S20>150 mJy. This allows us to estimate the extragalactic radio source counts at 95GHz by combining our observed 20-95GHz spectral-index distribution with the accurate 20GHz source counts measured in the AT20G survey. The resulting 95GHz source counts down to 80 mJy are significantly lower than those found by several previous studies. The main reason is that most radio sources with flat or rising spectra in the frequency range 5-20GHz show a spectral turnover between 20 and 95 GHz. As a result, there are fewer 95GHz sources (by almost a factor of two at 0.1 Jy) than would be predicted on the basis of extrapolation from the source populations seen in lower-frequency surveys. We also derive the predicted confusion noise in CMB surveys at 95GHz and find a value 20-30% lower than previous estimates. The 95GHz source population at the flux levels probed by this study is dominated by QSOs with a median redshift z~1. We find a correlation between optical magnitude and 95GHz flux density which suggests that many of the brightest 95 GHz sources are relativistically beamed, with both the optical and millimetre continuum significantly brightened by Doppler boosting.
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