Question: What to do when a sprint is finished early?

At the moment our Scrum team works off stories from the backlog, if the sprint is finished early.

What happens with stories taken from the backlog? Will the stories be added to the current Sprint ? If yes, what if these stories won't be finished in time. Is the Sprint failed then?

Answer: 
I'd encourage slack to be used for personal improvement. Sure, pull in stories from the backlog, but make sure you spend some time on yourselves: learn a new language, practice your craft with a kata, refactor some stuff, tweak, refine or write new tools to help you out, go and talk to a stakeholder, colleague or client, find out what your QA team does, take time to understand how your UX process works. 

There is a huge list of things you can do that will provide value to your business and yourself AND improve your velocity or the amount of quality value you provide that don't involve pulling things from a backlog, try those first.
Sprints do not address products but periods. If a sprint lasts two weeks, it cannot finish earlier.

If the products planned for the sprint have been finished in advance (I think your question goes in such direction), nice. But that doesn't mean that the sprint is finished. Get your backlog and follow on. 

In fact, a good scrum master would experience this situation almost half of the sprints. This should not be exceptional.

The end of the sprint is just a period to look back, understand what happened, assess the lessons learned, and make a new plan for the next sprint, based on this fresh experience. 

If your team advances quickly, you have two things to do. First, give them more work, because they need something to do. Second, and the most important move: **see how to reward them if they do deserve a reward** (naturally, a single sprint is not enough, but if they keep the pace along some time). If they're good, you don't want to separate them. You don't want the members to leave. You need the company to recognize the effort and react in consequence. You should make some noise about them being good. You need the stockholders to know them. A highly productive team is not easy to gather.