Welcome, hatish. I don't think I can make the connection for you, but here's what my Pequeño LaRousse Ilustrado gives as a definition for saeta:

saeta (lat. sagitta). Flecha, arma. || Manecilla del reloj || Brújula || Copla breve que se canta en las iglesias y en las procesiones.

The arrow, or weapon, definition is the first and the short religious song is the last. In between are the hands of a clock and a compass needle. I can see the connection between arrow, watch hands and compass needle but am puzzled by the short religious song definition. Perhaps, since the song is small and light, it is flung towards the heavens to more accurately reach God?