Really good play, dude! You made Annie look easy. I like watching streams and seeing what could be improved. I wonder if your items could have been better. Tell me what you think.
Like you said, Annie relies on snowballing a burst advantage into repeated kills. Buying more AP is nice, but early magic penetration is even more combat efficient, and a Dark Seal can offer ability power for waveclear while you buy it. So, the 1,700 gold of your first back could have been Boots, Aether Wisp, and Dark Seal plus a Control Ward and Health Potions.
You would have had safety against Fiddle from the ward, HP to trade to whittle down Ziggs from the Potions, more movement speed from Aether Wisp, and snowball potential from Dark Seal. From there, you could have stacked the Dark Seal off your snowball, getting free AP while buying Sorc Boots to reach your burst threshold faster.
From there, farming a Needlessly Large Rod would have been largely free, accomplished either by exploiting your lead or getting kills and plates. Spellbinder would have come just as your Dark Seal stacked, letting you follow up with Mejai’s to send your snowball out of control.
Finally, rather than opting for Luden’s Echo, you could have instead chosen Oblivion Orb, gaining even more MPEN off a fast 1,600 gold powerspike after which you could finish Deathcap.
With the money saved, your snowball could have finally been locked in with a Potion of Sorcery + Stopwatch buy. You would have been insanely strong yet also safe.
Hope you liked this! :)