Autosave saves automatically at the end of your turn, after you've finished doing everything, but before the computer has gone.  Basically, it's as if you had saved the game manually right before you hit End Turn.

So my question is (because this is what it looks like), are you saying that the computer takes an additional turn upon loading your save game, before you hit End Turn?  In other words, you load the game, and the computer-controlled units are not where they were?  Or are you saying that the computer-controlled units are actually moving upon loading the Autosave without you doing anything?

I haven't experienced any problems with the Autosaves myself.  I have mine set to save every turn, and I'll often load them after making mistakes since I'm still learning the game.

I know it would help the devs if you provided the autosave and quicksave files for comparison.