Attaching drywall to cracked/sagging plaster ceiling?
The plaster is sagging probably due to issues with some of the lathe boards above it.   Those need to be repaired.   

If you have access above this you could just knock these totally out and reframe with 2x4s.   This would probably be the quickest way.   

As far as mess...   You are thinking about this wrong.   Demoing might be a mess.   Sanding the mud for the new drywall will be 10 times worse for your house.   

My advice is knock out the ceiling.  (this will take minutes)   Then reframe and then add drywall.   This space will have to be totally emptied out.   If you are quick you can get drywall up with first coat of mud in day 1.   Day 2 you can add second - final set of mud (if you are really good).   This will take at least two days to dry.   Day 4 you are sanding.   That is best case scenario.   Room will have to be emptied and doorways will have to be tarped off from rest of the house.