You're balancing between two systems: your heart/lungs and your muscles.

If the legs are burning, shift to an easier gear and spin faster. This eases the stress on the legs and puts more stress on the heart/lungs.
If the heart/lungs are giving out, shift to a harder gear and mash away for a bit with your massive quads. This give you time to do some measured breathing and get your heart rate/breathing under control.

For getting breathing/heart under control, I like to hold my fist to the end of my nose to open up my nostrils (and wipe the inevitable dripping snot-sweat at the same time) as I find it helps to breathe in strongly through the nose a few times once in a while as opposed to gasping endlessly through my gaping mouth.

If both your legs and your heart/lungs are at the limit, you need to back off and recognize your limitations. Finish the hill as best you can and try to learn something about the particular turns and how you might take a more efficient line next time. Do this over and over and you will not only improve the fitness of both your physical systems, but you'll also know exactly when/where on the hill it's best to shift up/down and where there might be a flat(ter) spot where you might recover a little, etc.

In short: Keep riding.