In Blender, Shift-select the animated object ans its armature. So both are selected.
Go to File->Export, select FBX.
Next, browse to the folder where you want to export to, and enter an appropriate name.
Make sure to set the following configuration:
1. Check “Selected Objects”
2. At the right bottom, open the “Armature” settings and check “Only Deform Bones”, uncheck “Add Leaf Bones”
3. At the right top shift-select “Armature” and “Mesh”
Click “Export FBX”.
In Unity, Create a new folder under “Assets”. Give it an appropriate name (“Animations” in this example).
Now drag and drop your exported FBX file into this folder.
Done. From here on, you can setup and use the animation in Unity.