It seems that the holiness which belonged to Jesus Christ was not self-caused or intrinsic. Jehovah (the Father) evidently imparted holiness to Jesus, since the Son of God himself said:
"say ye of him, whom the Father sanctified and sent into the world, Thou blasphemest; because I said, I am the Son of God?" (John 10:36, KJV)
J.H. Thayer points out:
"So Christ is said by undergoing death to consecrate himself to God, whose will he in that way fulfills, John 17:19; God is said ἁγιάσαι Christ, i. e. to have selected him for his service (cf. ἀφορίζειν, Galatians 1:15) by having committed to him the office of Messiah, John 10:36, cf. Jeremiah 1:5; Sirach 36:12 (ἐξ αὐτῶν ἡγίασε, καί πρός αὐτόν ἤγγισεν, of his selection of men for the priesthood); Sirach 45:4 Sirach 49:7" (no. 37).
"God consecrates his own, incl. Christ J 10:36" (BAGD 8).
Jehovah also prepared a body for Christ (Hebrews 10:5ff), so based on Jn 10:36 and Hebrews, we might submit that the holiness of Christ was derivative, issuing forth from God in two primary ways:
1) He was set apart for God's service and 2) he was pure or sinless by virtue of his unique birth or the body which God prepared for him. I would also say that Jesus had to rely on his Father in order to be obedient under temptation (Hebrews 5:7-8). John recorded these utterances:
"And he that sent me is with me; he hath not left me alone; for I always do the things that are pleasing to him" (John 8:29).
"The Son can do nothing of himself, but what he seeth the Father doing" (John 5:19). Christ became and remained holy through the eternal spirit which emanates from Jehovah.