We propose a novel approach for automatic extraction (instance segmentation) of fibers from low resolution 3D X-ray computed tomography scans of short glass fiber reinforced polymers. We have designed a 3D instance segmentation architecture built upon a deep fully convolutional network for semantic segmentation with an extra output for embedding learning. We show that the embedding learning is capable of learning a mapping of voxels to an embedded space in which a standard clustering algorithm can be used to distinguish between different instances of an object in a volume. In addition, we discuss a merging post-processing method which makes it possible to process volumes of any size. The proposed 3D instance segmentation network together with our merging algorithm is the first known to authors knowledge procedure that produces results good enough, that they can be used for further analysis of low resolution fiber composites CT scans.