IMHO, most "magic" associated with the Elves in Middle-Earth has a connection with the Valar, or with song & recitation:
The Girdle of Melian: The fence of enchantment set around Thingol's kingdom of Doriath by Melian his Queen, preventing entry into his land without his will and consent. Melian was a Maia.
The Silmarils: crafted by by Fëanor in Valinor, after being taught the craft by Aulë.
The Elven Rings: crafted by Celebrimbor, the son of Curufin and grandson of Fëanor (and thus by extension the craftsmanship comes from Aulë).
The Mirror of Galadriel: the origin of the mirror is not detailed, but The Silmarillion states that the Noldor took with them many items from Valinor when they departed. Since the Noldor were favored by and were closest to Aulë, it's not unreasonable to assume he had some doing in its making.
The rising of the waters of Bruinen: according to The Silmarillion, all water is under the governance of Ulmo:
For all seas, lakes, rivers, fountains and springs are in his government; so that the Elves say that the spirit of Ulmo runs in all the veins of the world.
Song & Recitation:
The singing of Lúthien to defeat Melkor. Lúthien of course is half Maia as well.
The singing of Tom Bombadil to control the trees in the Old Forest.
Sam's calling upon Elbereth in Cirith Ungol, holding the Light of Eärendil:
A Elbereth Gilthoniel, o menel palan-diriel, le nallon sí
di-nguruthos! A tiro nin, Fanuilos!
(O Elbereth Starkindler, from heaven gazing far, to thee I cry now beneath the shadow of death! O look towards me, Everwhite)
Also, the Light of Eärendil comes from the Silmarils.
Considering the world actually came into being through music according to the canon, it is not surprising that music and recitation would have magical properties.
Even in the movies, though not following the books, when Arwen calls upon Bruinen (which incidentally stills sends shivers up my spine):
Nîn o Chithaeglir lasto beth daer; Rimmo nîn Bruinen dan in Ulaer!
(Waters of the Misty Mountains, listen to the great word;
flow waters of Loudwater (= Bruinen) against the Ringwraiths!)