I figured out a way to do this using the QGIS OpenLayers Plugin and creating a TMS viewer for the tiles. Then I added the new viewer to the list of layers in the plugin (openlayers_plugin.py).
The viewer is not ideal. It doesn't refresh the image when zooming, only when panning, and could do with many improvements (not sure if this is a factor related to the viewer or the fact that it's using a different OpenLayers.js file), but it shows that SuperOverlay tiles can be read in QGIS.
My attempt at a viewer (modified from the openlayers.html file generated by gdal2tiles):
https://github.com/rudivs/qgis-openlayers-plugin/blob/master/openlayers/html/tms_layer.html