I haven't been reading up on the thing, but from both ADS-B data and people talking about the video of the crash, it sounds like it was pitching up and down alternately and in the video, at least some of the flight surfaces appeared to be disabled, were flopping around with gravity.

That airliner that crashed on a runway in Sioux City some decades back exhibited similar symptoms. Basically, after an engine tore itself apart, they had all three redundant hydraulic systems be penetrated, so they had no control of the flight surfaces. They managed to figure out that they could use engine power to the remaining engines alone -- which didn't use the hydraulic systems -- for sufficient control to crudely fly the airplane. The problem was that they had this pitching up and down that happened -- they managed to get it to the airport and line it up with the runway, but right as it was landing, it did one of those pitching motions, and so it crashed, which it sounds like is pretty much the same thing as what the Azerbaijan Airlines airplane here did. In both crashes, about half the people aboard died, half lived. I believe that at the time, it was considered to be quite a feat of piloting to figure all this out and manage to save half of the people.

kagis

United Airlines Flight 232.

https://en.wikipedia.org/wiki/United_Airlines_Flight_232

So if the missile ruptured all of the the hydraulic systems on Azerbaijan Airlines Flight 8243 -- which is what happened with another SAM attack on an aircraft, the 2003 Baghdad DHL shootdown -- that might be a route for the same phenomenon to occur.

UA232 was a triple-engine aircraft, and lost an engine in the failure that severed the hydraulics, so had two remaining engines, which is the minimum you'd need to fly without control surfaces. Azerbaijan Airlines Flight 8243 had only two engines to start with, so if they were controlling the aircraft with engine power alone, all the engines would have had to be functioning up until the end, or the aircraft would have needed to have have gone down right after being struck.

EDIT: It looks like someone's already updated the UA232 Wikipedia page linking to Azerbaijan Airlines Flight 8243 as a similar incident and saying that it also probably suffered total hydraulic control system failure, so it's not just me speculating along those lines.