Can't I just club Azelf upside its head and shove it in a bag?

I know legendary pokemon aren't supposed to be easy to catch. I get that. But when its HP is a single stack of red pixels and its got paralysis. I'm feeling a bit defeated after it ate 30 Ultra Balls and 6 Timer Balls. (at which point I was distracted and didn't realize I'd accidentally told my level 34 Chansey to slap it instead of healing itself.

Wait 'til you start trying to catch the legendary runners, who'll flee just like Abra without Mean Look, Arena Trap, or something similar to prevent them! ;-)

Note the parts I put in bold from your quote. My advice is to put them to Sleep whenever possible, since Paralysis is only 75% as effective as Sleep in terms of how much it helps you catch stuff. Also, use Dusk Balls (since you're in a cave), which are twice as effective as Ultra Balls. This literally doubles your catch rate.

Roke Tanuki said:

Is there some magic formula for catching them? Some magical skill that can be pull't-out-the-ass? Can I just walk up and crack Azelf across the back of the skull then shove it in a backpack?

There's a formula, but it's not magic; it's Satanic. Here it is on Bulbapedia.

I've done the math, and your percentage chance to catch a sleeping Azelf with 1 HP left, using a Dusk Ball, is ~9%. A paralyzed Azelf, using Ultra Balls, probably has about a 3% catch rate by comparison. See, every Poke has a base "catch rate". In the case of Azelf, Mesprit, and almost all legendaries, the catch rate is... 3. That's compared to Starly, for example, which has a catch rate of 255. ...Yeah, it's like that. Their remaining HP, max HP, status abnormalities, and the type of ball you're using all modify this basic catch rate.

The key is to stock up on balls and keep using them over and over. And, if you use a whole lot of them, start tossing your Timer Balls, too.

It took me ~40 Dusk Balls (4x catch rate) to catch Azelf with 1 HP while it was sleeping. OTOH, I caught Zapdos on the third toss in FireRed with an Ultra Ball.

Rone Tanuki said:

Can you sleep a character that's already suffering Paralysis? Some advice and strategies would be wonderful.

Tanks ahead of time for the advice. =)

No, a Poke can only have one status abnormality at a time (except for confusion and stat reductions). Basically, only one of any stat abnormality that puts an icon on their status bar.

You're welcome!

EDIT: And I have some more to add. Basically, it's this: I, too, got frustrated when I first started catching legendaries again. The key, though, is to be aware of how to maximize your catch rate, and accept the fact that: 1.) You might get lucky and catch it quickly, or 2.) It might take quite a bit of patience. You may even have to restart and try again (although you can't do that with runners, obviously).

With Azelf, I can tell you that if you get its HP down to 1 (False Swipe), put it to sleep, and throw a Dusk Ball, you will have ~9% chance to catch it. This doesn't mean you'll catch it within eleven tosses, though. It could be on the first toss, or it could take twenty-five or thirty (or, if you're very unlucky, forty) tosses. But statistics will win you victory eventually, almost always sooner rather than later.

The worst ones to catch are the ones who run through the grass, appear randomly, can escape if you don't trap them, and can use Struggle or a similar move to commit suicide. Azelf, by comparison, is a piece of cake.