International airspace is given over to control by different agencies, with the dividing lines set by ICAO (International Civil Aeronautics Organization).
Around Hawaii (within about 300 miles or so) it is US airspace, just like in the mainland. The rest of it is international airpace, controlled by Oakland Oceanic control. The surprising thing is that Oakland controls airspace most of the way across the Pacific.
This link includes a small chart of the various airspace control areas in the Pacific. http://www.airservicesaustralia.com/pilotcentre/training/RVSM/01over/01over02a1.htm
One other interesting thing is that we don't talk directly to Oakland control while flying over the Pacific. A private radio company called AIRINC acts as a middleman between us and Oakland, and also between us and our company dispatchers. We talk to them on the HF radio, and they relay the information about when we pass waypoints, and position reports and requests for altitude changes to the controllers. Then AIRINC relays the responses back to us. It's a little bit slow and clumsy, but for now that's the way it is done.
One smooth landing is skill. Two in a row is luck. Three in a row and someone is lying.