No they do not!!!
If my car only drove in reverse and started 80% of the time would you consider that “working”?
Certainly not if I was trying to convince you to buy the car at full market price. You wouldn’t consider it working because it has partial functionality paired with unreliability. And that describes the PS4 controller connected to a PS3 console.
Unfortunately there’s a plethora of videos on YouTube hosted by snot-nosed teens demonstrating PS4 controllers “working” on the PS3 console. They’re not faking it they really are controlling the games with the PS4 controller but that’s not the whole story.
What they typically forget to mention is that the PS button doesn’t work. You’re probably thinking that just means you can’t wake the PlayStation with the controller. That’s no big deal though right? No oddly that’s the only thing the PS button can do. Which suggests that the button can communicate with the console but after that, possibly deliberately, it has no functionality. Without the PS button you can’t enter the controller settings, you can’t turn the console off, you can’t exit a game once you’ve entered it, and there’s no guarantee all the other controller buttons will work as expected once you’re in a game. Some games won’t let you in at all.
For example “The Orange Box” Valve’s collection of Half Life, Portal and Team Fortress 2 on a single disk actually checks what controller you’re using when the game loads. Seems like a strange thing to check. I don’t know why they care? Anyway if the game doesn’t detect a standard PS3 controller you can’t progress. This is ironic considering one of the best ways of connecting a PS4 controller to PC is by using the Steam platform by Valve, there’s no native way to do it via Windows. If you’re thinking that’s fine I’ll progress to the in-game levels with a PS3 controller and then reassign that controller as the second controller, well that doesn’t work. The PS4 controller won’t take over as the first controller and you can’t change it manually because . . . you guessed it, you can’t bring up the controller settings on the PS4 controller because the PS button doesn’t work.
So in summary if you want to use a PS4 controller you’ll likely need to have at least a partially functioning PS3 controller to use with it, but even then some games may not work at all. If you’ve no controller and need one for the PS3 (they don’t make them anymore) you’ll have to buy used or a high quality clone (and that’s a whole other mess).
Maybe someday Sony will be cool and release a software update for the PS3 console that will allow the PS4 controller to work with it. Comment below what you think that chances of that happening are.