Good quality blender animations with bones become poor quality legacy animation in unity?

I've taken the blender model of a human hand from here http://www.libhand.org/. I animated a hand grasp, and it looks perfect in blender. This model has an armature and bones.

I exported the model as an fbx and imported that fbx into unity. It looks good when i'm in edit mode (no animation), I see the textures and the animation does have the general motion I intended it to have. But when I see the preview game mode, the mesh and textures are deformed as I've shown in the following screenshots.

This is a blender screenshot

This is a unity screenshot

Is this a problem with armatures and bones? I'd like to keep animations in blender but I don't know what to do as of now.