For those who didn't know what this means/how I solved it:
Select the Animation in the Action Editor (that is: Dope Sheet View, there Mode: Action Editor) and then switch to the NLA Track Editor.
There you'll see your action's name and an double arrow pointing downwards. If you click it, a new NLA Strip will be generated which contains your Action. Strange shit, but it worked
The downside is that playing the animation distorts my mesh into some spikey-looking mess. Not sure what's wrong there?