A good teleconverter or extender works better than cropping if the lens can outresolve the resolution of the sensor by an amount greater than the degradation of the TC and the resulting loss in lens speed is not a determining factor in the viability of the shot.
It depends entirely upon the teleconverter in question, the lens in question, and the full frame and APS-C cameras in question. The same is true when comparing a full frame camera + 1.4x to an APS-C camera when both use the same lens. Optical quality is never better than the weakest link in the equation. Adding a low grade TC to a high grade lens and camera should be expected to degrade image quality worse than cropping with a bare lens. But not all extenders are optically inferior to high grade lenses.
I've seen in-depth analysis that has gone one way with a particular camera/converter/lens combo and other comparisons with different cameras/converters/lenses that swung the other way. There are way too many variables from one camera/teleconverter/lens combination to the next to usually be able to say for sure one way or another which is best.
Consider just one such variable: the needed resolution of the final image. If the final use of the image only requires a resolution of, say, 2400x1600 pixels (3.8MP) then it doesn't matter at all if you crop a 22MP FF image down to an 8.6MP image to get the same angle of view as a 22MP APS-C camera would deliver with the same lens. If, on the other hand, you need all 22MP then the TC on the FF or the APS-C camera with the same bare lens becomes much more attractive.
Let's look at another variable: diffraction. If you need to shoot at f/11 which camera will have an advantage? The FF camera with a DLA of f/10.1 or the APS-C camera with a DLA of f/6.6? Using a smaller sensor does not reduce the size of the wavelength of light at a specific frequency.
Along with the transmission loss and potential aberrations due to the additional glass a teleconverter places in the optical path one needs to also consider the differences in efficiency of the two sensors being compared. Due to the way the ratio of linear edges to area increases as a pixel gets smaller, a sensor with larger pixels and otherwise the same technology will be more efficient. The properties of light in terms of wavelength size don't change with the size of the sensor. Even with so-called "gapless sensors", smaller pixels mean a lower percentage of the light falling on a sensor actually makes it to the bottom of a pixel well and is measured by the sensor.
When using an extender or teleconverter with a telephoto zoom lens, one should also consider the positive contribution a TC can have on geometric distortion. Most telephoto zoom lenses demonstrate mild pincushion distortion at the long end (which is where you'd generally use a zoom lens with a TC). Most quality TCs introduce a little barrel distortion that counters the pincushion to one degree or another.
Then there are considerations not directly related to pure optical image quality. If you are shooting action or sports, for example, it doesn't matter how great the image quality you get is if your AF system is too slow or too inaccurate to nail focus on the intended subject at the critical moment. With lower end lenses (i.e. narrower apertures) and teleconverters, focusing tends to be slower and less accurate with a TC if AF even works at all. Due to the narrower baseline allowed by the size of an APS-C lightbox and mirror, similarly designed PDAF systems, which measure light coming from opposite sides of the lens, will perform better in terms of accuracy and shot-to-shot consistency in a FF camera that allows a wider baseline. With premium lenses and the best TCs, focusing speed takes much less of a hit and the advantages in terms of accuracy and shot-to-shot consistency of the wider AF baseline allowed with a larger sensor and mirror often outweigh the difference in speed.
Having said all of that: in general when the best cameras are coupled with the best lenses the best teleconverters will usually give better image quality of static subjects shot from a tripod mounted camera than using the same camera without a TC and then cropping away. The Canon EF 70-200mm f/2.8 L IS II + Canon EF2X III yields better IQ than the EF 400mm f/5.6. Not by a whole lot, but it is better. The combo is also more expensive. However, even with the same camera/TC/lens, if the stop of light lost with a 1.4X is more crucial than the loss of resolution from cropping in such a case cropping might be the better the way to go.
Also generally speaking, a crappy lens coupled with a crappy TC will almost always look worse than just cropping with the same lens or using an APS-C camera instead of a full frame camera. Remember what we said at the beginning:
A good teleconverter or extender works better than cropping if the lens can outresolve the resolution of the sensor by an amount greater than the degradation of the TC and the resulting loss in lens speed is not a determining factor in the viability of the shot.