B767 Captain reported cabin pressurization malfunction during cruise flight. Flight crew donned oxygen masks; descended to safe altitude; and continued to destination.

Narrative

Approximately 15 minutes after we leveled off at our cruising altitude at FL400 the CABIN ALTITUDE warning EICAS displayed with the siren sounding and the two cabin altitude warning lights illuminating. I glanced at the cabin altitude control panel and saw the cabin altitude indicator reflecting approximately 10;100 feet. The First Officer and I along with our three jumpseaters immediately put on our oxygen masks and we accomplished the CABIN ALTITUDE QRH procedure. I notified ATC that we had a pressurization problem and we needed an immediate descent. ATC gave us a descent initially to an altitude in the low 30s/upper 20s so I told ATC we needed 10;000 and [requested priority handling] because I didn't want to have to level off before reaching 10;000. After we reached 10;000 feet I looked at where the cabin altitude indicator was on the control panel and saw it was just under 10;000 feet. The cabin continued to descend and after several minutes it was at approximately 3;000 feet. The First Officer and I discussed our options of continuing to ZZZ [airport] or diverting. We decided to continue for the moment because the situation was stabilized with the cabin at approximately 3;000 feet and the FMC reflected us landing with 14.3 and our bingo fuel to our ZZZ1 [airport]; the alternate; was 13.7. After several minutes at 10;000 feet we decided to climb back up to 16;000 feet because the cabin altitude was holding steady at approximately 3;000 feet. At that point; I sought inputs from our jumpseaters (two captains and one first officer). We discussed our options and the need to notify flight control. Soon there after; I contacted flight control and maintenance to get their inputs. Flight control; Maintenance and I agreed it would be OK to climb in the 20s as long as I was comfortable with it and the cabin remained stable below 10;000 which it was at the time. Flight control gave me new fuel burn numbers from our position southeast of ZZZ2 [airport] with ZZZ3 as the new alternate. Shortly thereafter we ended up climbing to 28;000 feet and remained there until the start of our arrival into ZZZ. The cabin did so very slowly climb back up to approximately 6;500 before our descent into ZZZ. We landed safely in ZZZ with 17;200 pounds of fuel with no further issues.