B737-700 Captain reported a Cabin Altitude Warning Horn and a Cabin Altitude of 9;500 feet in cruise. The Captain requested priority handling and directed the First Officer to descend to 10;000 feet. The flight crew elected to continue to destination airport at 10;000 feet and make a precautionary landing.

Narrative

While flying at FL 380 from ZZZ1 to ZZZ; we experienced a Cabin Altitude Warning Horn and noticed our Cabin Altitude was displaying 9;500 ft. We accomplished the immediate actions items for this event and I called for the checklist. While accomplishing the Cabin Altitude Checklist; we noted that the Cabin Altitude was continuing to rise to 10;000 ft.; so I requested priority handling and directed the F/O (First Officer) to start an immediate descent. I informed ATC that we were starting this immediate descent to 10;000 ft. MSL. The Cabin Altitude started to decrease as we descended. After leveling off at 10;000 ft.; I asked to proceed direct to ZZZ. After resetting our cruise altitude in the FMC; I saw that our arrival fuel was projected to be 7.5. I contacted Dispatch via ACARS to inform him of my plan to continue to ZZZ at 10;000 ft. I also contacted the Dispatcher on SATCOM. I informed the Flight Attendants of the pressurization problem during the descent. I contacted them again during cruise at 10;000 ft. to ask if the masks had dropped; and if any of the Passengers noticed anything strange to determine whether or not to make a PA. They told me the masks did not drop and none of the Passengers noticed anything; so I decided a PA was not warranted. We proceeded direct to ZZZ and landed with no further problems.There were several issues/mistakes which the FO and I discussed after the fact. We skipped the QRC and went straight to the QRH. We should have complied with the QRH step to activate the Passenger Oxygen System. Even though the cabin altitude was not climbing rapidly; the fact that I felt the need for an immediate descent to control Cabin Altitude means I did not feel the Cabin Altitude was under control. I should have considered stopping the descent before reaching 10;000 ft. when Cabin Altitude was within limits. We forgot to accomplish QRH deferred items. One additional note: We flew this aircraft from ZZZ2 to ZZZ1 at FL 320 without noticing a cabin altitude problem.