Ancient Church Father, Cyril of Jerusalem

Cyril of Jerusalem details how ointment or oil was "symbolically applied to thy forehead, and thy other organs of sense" and that the "ears, nostrils, and breast were each to be anointed." Cyril states that the "ointment is the seal of the covenants" of baptism and God’s promises to the Christian who is anointed. Cyril taught that being "anointed with the Holy anointing oil [Chrism] of God" was the sign of a Christian (Christos means "anointed"), and a physical representation of having the Gift of the Holy Spirit (Holy Ghost). He say's; "Having been counted worthy of this Holy Chrism, ye are called Christians, verifying the name also by your new birth. For before you were deemed worthy of this grace, ye had properly no right to this title, but were advancing on your way towards being Christians."

"When one is baptized one becomes a Christian," writes Cyril, "exactly as in Egypt by the same rite one becomes an Osiris" (Patrologiae Latinae 12:1031), that is, by initiation into immortality. The baptism in question is a washing rather than a baptism, since it is not by immersion. According to Cyril, this is followed by an anointing, making every candidate, as it were, a messiah. The anointing of the brow, face, ears, nose, breast, etc., represents "the clothing of the candidate in the protective panoply of the Holy Spirit," which however does not hinder the initiate from receiving a real garment on the occasion(CWHN 4:364).
Furthermore, according to Cyril, the candidate was reminded that the whole ordinance is "in imitation of the sufferings of Christ," in which "we suffer without pain by mere imitation his receiving of the nails in his hands and feet: the antitype of Christ's sufferings" (Patrologiae Graecae 33:1081).
The Jews once taught that Michael and Gabriel will lead all the sinners up out of the lower world: "they will wash and anoint them, healing them of their wounds of hell, and clothe them with beautiful pure garments and bring them into the presence of God" (R. Akiba, cited in CWHN 4:364). Meaning and Functions of Temples by Hugh W. Nibley.

See Cyril of Jerusalem’s Lectures on the Ordinances, an early Christian document circa 347 ce. The document meshes nicely with the book of Revelation and also describes early Christians as participating in a ritual where they were washed, anointed, and clothed with special garments. The document also speaks about prayer circles and a ritual that starts with a description of the creation of man and the Garden of Eden, and then moves to a world full of temptation. The Lectures concludes with the admonition, “Keep these traditions inviolate, and see that you do not stumble.” Cyril’s Lectures are yet another proof of early Christians’ ancient temple connections.

"Neither shall he go out of the sanctuary, nor profane the sanctuary of his God; for the crown of the anointing oil of his God is upon him: I am the Lord." (Leviticus 21:12)

Parallels in Judaism and early Christianity

Multiple early Christian documents discuss the ordinance of "chrism" which is nearly identical to the anointing ordinances in Latter-day Saint temples. The most detailed version of the practice is by Cyril.
The text also claims that all true, believing Christians received the ordinance: "We are called Christians, because we are anointed with the oil of God."

Also see:
The “Mark” of Anointing
Cyril of Jerusalem
On Chrism by Cyril of Jerusalem