I did a bit of digging, in case there was (as with some of the other lyrics) some original Tolkieny bits in the odd phenom.

http://tolkiengateway.net/wiki/Moon notes that the moon did, in fact, originally rise in the west:

“After the Darkening of Valinor and the destruction of the Two Trees, Telperion the White Tree bore one last Flower of Silver before its end. Aulë and his people made a vessel to carry to the silver flower aloft, and Tilion, one of the hunters of Oromë; was granted the task of steering the new Moon through the sky.

“Tilion guided his charge up into the western skies just as the Noldor were returning into Middle-earth, and so marked the beginning the First Age. The Moon first rose above Valinor in the far West of the World, but Varda came to change this arrangement, so that the Moon would pass beneath the World, and arise in the east instead, as it does to this day.”

I can’t quite parse the song (http://tolkiengateway.net/wiki/Into_the_West) into fitting that scenario, unless our PoV in the song is supposed to be back on the Grey Havens:

“What can you see On the horizon? Why do the white gulls call? Across the sea A pale moon rises The ships have come To carry you home”

Clearly the ships come to the western shores of Middle-Earth, “across the sea” from the Uttermost West. I think you could argue for an elided “[Here we are] across the sea [from that horizon, and here] a pale moon rises [as] the ships have come to carry you home.”

Or maybe it was just, um, poetic license.