We meet again o/
Adding onto this: While playing Anivia, I noticed some players would be unusually forward and agressive. I guess what probably happened is they heard that Anivia's Q is hard to land, so they thought they didn't have to do anything special or try to dodge it. But the more aggressivle you're playing into Anivia, the easier it is for her to hit you with it. So if her Q is being such a problem for you, you're probably playing too aggressively. She's really good at punishing aggressive play, and even strong early game champions can struggle to so much as pop a good Anivia's passive.
The two best things you can do against Anivia is learn Lane Control and Dynamics, and fuck with the rest of her team.
Early on if you shove the wave into her Tower, you can make Anivia miss CS. lowering her CS delays her mana items, which makes it harder for her to clear waves and causes this painful cascade effect that keeps delaying her next stage.
After 6, you have to wait until after Anivia clears the wave to put your Pool down. She clears waves much faster then you do (She's a monster at it, really), so if you put your pool down first she'll just kill your minions and then hers will just walk out of your pool. That really sucks, and you need to watch out for all champs that can clear waves fast like that.
Anivia has always been a pretty team oriented champion, and that's even more true now after her rework. So, if you can shut her teammates down, you can really slow her down too. If she's got a fed Tank, you also really need to look out for them enabling eachother. Basically what happens is one of them lands their CC and, then the other one does and Anivia Ults and walls off their escape... If that happens, it can be really nasty to deal with. Roaming/teleporting and shutting down her team is your best option for keeping Anivia from going off, especially as a control mage like Morg. Teleport is better then combat summoners in this matchup, because it's not really like you're going to fight Anivia much anyway. Maybe cleanse if she has a tank(without a knockup.......)
It's been awhile since I've played, but I used to main both of these champs so hope it helps o/