Is it safe to let my baby fall asleep in the swing? Or do I have to get her out and put her in the crib after?

Don't worry, its fine. When my son was a new born thats the only place he would sleep. He had really bad gas issues and the motion would sooth is tummy. I don't suggest it for months on end, eventually you do need to have her sleeping in her crib, and falling asleep in her crib. im guessing she is a newborn though, so don't worry:)