I have dealt with this before and there isn't an easy fix. I remedied the situation this way;
1) First, I scraped the ceiling trying to remove any loose paint. Oddly not much would scrape off even though it easily came off with the roller. I then did a quick cleaning with fresh water and a sponge to remove any dust. I noticed that the ceiling absorbed some of the water so I decided to proceed with the next step.
2) After allowing the ceiling a couple of hours to dry, I applied a bonding primer. I used Peel Stop. This is a thin latex emulsion that can penetrate into the pores of some paints and glue it to the surface. I used a new 1/2 inch lamb's skin roller cover and the least pressure possible when applying. Some of the paint did come off but not much. It isn't possible to apply Peel Stop thickly, so I planned on 2 coats. Wait 2 hours before applying a second coat. I allowed the primer to dry overnight before going to the next step.
3) I applied Kilz to the areas needing patching; regular drywall mud was used to smooth these areas with their surroundings. The patched areas where reprimed after the drywall mud dried and was sanded smooth.
4) Now it was time to paint. I added Flotrol to the paint to help with easier application and reduce the drag on the roller, plus using the lightest pressure possible helps. Two coats of paint was applied.
The ceiling looked good and the customer was happy. I never received a call back for additional repairs so I assume it worked.