We report the first astrophysical application of the technique of wide-field interplanetary scintillation (IPS) with the Murchison Widefield Array (MWA). This powerful technique allows us to identify and measure sub-arcsecond compact components in low-frequency radio sources across large areas of sky without the need for long-baseline interferometry or ionospheric calibration. We present the results of a 5-min observation of a 30x30deg^2^ MWA field at 162MHz with 0.5s time resolution. Of the 2550 continuum sources detected in this field, 302 (12 per cent) show rapid fluctuations caused by IPS. We find that at least 32 per cent of bright low-frequency radio sources contain a sub-arcsecond compact component that contributes over 40 per cent of the total flux density. Perhaps surprisingly, peaked-spectrum radio sources are the dominant population among the strongly scintillating, low-frequency sources in our sample. While gamma-ray active galactic nuclei are generally compact, flat-spectrum radio sources at higher frequencies (162MHz), the properties of many of the Fermi blazars in our field are consistent with a compact component embedded within more extended low-frequency emission. The detection of a known pulsar in our field shows that the wide-field IPS technique is at the threshold of sensitivity needed to detect new pulsars using image plane analysis, and scaling the current MWA sensitivity to that expected for SKA-low implies that large IPS-based pulsar searches will be feasible with SKA. Calibration strategies for the SKA require a better knowledge of the space density of compact sources at low radio frequencies, which IPS observations can now provide.