As far as power Akatsuki are hopelessly outmatched.
Episode 4 and 5 involved John Stewart destroying a planet, by
accident. Granted it turned out he didn't actually do it but nobody at the trial questioned that he
could accidentally turn a planet into an asteroid field without trying and that would have been a pretty good defense in the trial. Superman punched Darkseid through seven skyscrapers in the world of cardboard scene. Akatsuki are quite strong but not Justice League strong.
Granted Vigilante or The Creeper aren't going to make as good a showing but they aren't needed anyway. This makes the Justice League rather an OCP for the Akatsuki, all the ones who are used to solving problems by force are going to go down almost immediately because they're so outclassed in the strength department.
Superman has a track record of seeing stuff underground and one can reasonably extrapolate that Supergirl can do it too. This means stuff like their usual methods of hiding underground (without genjutsu) are going to fail.
Martian Manhunter brings telepathy to the table and is strong enough for the job but weak to fire, so he'd have to be aimed only at Akatsuki who aren't likely to throw fire around and there aren't that many of them so his use will be limited. He's probably going to be assigned to capture Hidan.
Batman of course will figure out a strategy and somehow his hitting a ninja with a batarang will be as useful in the fight as Wonder Woman throwing somebody five miles into a mountain.
We can reasonably discount any Akatsuki who relies on brute force and any who rely on hiding without special genjutsu to protect them. That means Deidara, Hidan, Kakuzu, Kisame, and Sasori can pretty much be taken out without much difficulty. Konan will probably go down as well, turning into paper isn't a good enough trick to evade the kind of speed the Flash can use to gather her up, nor will it let her evade just being grabbed by a Green Lantern projecting a bubble.
That leaves Itachi, Pein, Orochimaru (maybe), Tobi, and the Zetsus as the main tevent. Zetsu seems to be supernaturally stealthy, Pein's actually got enough brute force to be a problem, and Itachi, Orochimaru, and Tobi bring genjutsu and space-time jutsu to the table.
Figuring out which hero can resist which genjutsu is going to be a hard question. Batman's probably going to shine here as he'll spot minute flaws in the genjutsu and somehow counter it. This is probably where he gets to hit somebody with a surprise batarang coated in sedative or such. Superman and Wonder Woman have shown themselves vulnerable to illusion. Martian Manhunter's telepathy may help.
Pein's likely to go down once either a Kryptonian figures out where his real body is or Martian Manhunter senses his. He'll wreck some havoc before that but ultimately he's still bringing brute force as his main MO.
Itachi has a really nice assortment of powers that are going to be hard for the Justice League to counter. Their best chance is to have The Flash speed blitz him while he's standing in the open before he can react. Tobi is pretty much the same except even more slippery and like just to teleport off if things don't go his way. The problem is that the Justice League aren't likely to open with those tactics and probably won't speed blitz first. Somehow I see Batman being the one to confront Itachi in a showdown.
Orochimaru is also tricky but may not be involved and he's probably on the weaker end, basically Itachi light with more biology going on.
Zetsu has a fair chance of escaping and causing the Justice League to lose. He's really sneaky and has no tendency to even try to confront an enemy he can avoid.
Overall I'd say the Justice League can take all Akatsuki's heavy hitters but would take minor losses against Pein, and potentially heavy enough losses against Itachi and Tobi to have to withdraw though they have a chance of taking them down as well. Zetsu and Tobi are going to escape more often than they're caught which is a lose condition for the Justice League.
Ultimately it's the Justice League's fight to lose. They have enough power to take it but their tactics and tendency to take enemies alive will work against them. Even knowing the enemy's jutsu ahead of time, it's not in character for them to have The Flash streak through in the first millisecond and hit all targets with a Phantom Zone projector or similar and that would be needed to keep Zetsu from slipping away.