My guess would be that you can listen to the 'measure-end' event on the dijit, which will give you access to the result of the measurement and a geometry.
From here, you could take that geometry and create your own graphic from it. You could use the measurement result (also from the measure-end event) as an attribute on that graphic. This graphic can then be added to a feature layer (probably a polyline feature layer) in order to persist it. You can add a label class to the feature service backing up that feature layer that looks at the attribute that represents the measurement result and applies it as a label.