It means the night has gone far faster than what you expected and you almost didn't notice it go by. An example is you were working on a project all night and didn't notice the hours go by because you were working so hard. Suddenly you realise it is almost morning. You say to yourself (or someone with you), 'This night got away from me'. You are saying the hours flew by without you knowing and before you knew it the night was almost over. Variations of this expression are fairly common.

I have never heard this expression before. But, with the help of Google Translate, I'd conjecture that it's something like "I missed this night. This night just disappeared from me."