It seems that the concept of the Christian minister being a “priest” took place in the thought of Cyprian as mentioned in a letter (63) written approximately 253 A.D:

This letter is also noteworthy for it contains, in the estimation of the incisive Congregationalist theologian P. T. Forsyth (1848–1921), “an absolutely unscriptural change.” After linking the biblical affirmation about the offering of Christ, the high priest of God, “as a sacrifice to the Father” with His command to His disciples to celebrate the Lord’s Supper in His remembrance, Cyprian concludes that Jesus is asking His disciples to do exactly as He did. This means that the one presiding at the Eucharist “imitates that which Christ did,” when he “offers a true and full sacrifice in the Church to God the Father.” In making this exegetical move, Cyprian became, according to Forsyth, “the chief culprit in effecting the change from a sacrificium laudis by the Church to a sacrificium propitiatorium by the priest.”

Whether or not Forsyth is right to designate Cyprian as the “chief culprit” in this regard is moot. What Cyprian’s words do reveal is that earlier usage of the term “sacrifice” with regard to the Eucharist—as found, for example, in the exegesis of Mal 1:11 by Justin and Irenaeus, an interpretation with which Cyprian generally agrees—has probably helped prepare the way for the significant change reflected in this letter of Cyprian. Whereas Justin and Irenaeus, however, saw the people of God corporately offering up the sacrifice of the Eucharist in purity of heart, Cyprian identifies the bishop or minister as the one who is uniquely called to do this and who, in this aspect of his ministry, imitates the high priestly sacrifice of Christ Himself. Fundamental to this shift in focus is Cyprian’s use of the term “priest” (sacerdos) as a description of the one presiding at the Eucharist. Prior to Cyprian, this term is never used to designate Christian ministry per se, but Cyprian, as in the letter under consideration, continually calls the one presiding at the Eucharist, whether bishop or elder, a sacerdos. And just as there are types of Christ’s passion in the history of God’s people preceding the incarnation, as Cyprian outlines in this letter, so in the history of the church since the death and resurrection of Christ there are priests who imitate Christ’s priesthood and who are vehicles for His presence in the church’s worship.

Schreiner, Thomas R.; Crawford, Matthew R (2011-01-01). The Lord’s Supper; Michael A.G. Haykin, “A Glorious Inebriation: Euchartic Thought and Piety in the Patristic Era” (New American Commentary Studies in Bible & Theology) (p. 111-112). B&H Publishing. Kindle Edition.

Here is the section of Forsyth’s The Church and Sacraments, which Haykin references:

 Then, thirdly, we have the long deep error to which Loofs chiefly alludes—not only the association with the Eucharist of the idea of sacrifice, but the treating of it as a sacrifice (in a cult, which the ‘breaking of bread’ was not). It is easy to see how the idea would creep in.We begin rightly enough by treating the occasion as a sacrifice of praise. The eucharista was an offering of thanksgiving.We have it so in the Didache. It was the fruit, the calves, of the lips. In the New Testament prayer is figuratively spoken of in this way. And well-doing and fellowship (Revelation 13:16) are sacrifices that delight God.The gifts of love brought at the Supper for the poor could easily he so described. But the transfer of the ea to the whole rite was of the most fatal import and consequence.There is no such description of it in the New Testament, though it came in soon after. It is proper enough in the Act to present before God the finished sacrifice of Christ as his gift to us, and therefore the best sacrifice we have to give. But when we re-enact the sacrifice of Christ, when we repeat the Cross instead of pleading it, we not only cause man to offer up a Christ who alone could offer himself (as Judas forced his hand), but we hide the ruling idea of Christ in the Church’s midst, offering himself and his finished work afresh to his own.The more the act was removed from the community to the official, so much the more deadly was its transfer to be a real sacrifice or a repetition of Christ’s sacrifice, and the transfer of the administrator to be a priest.‘As the sacrificial idea passed from its innocent use it took up seriously the priestly idea.Arid as the priestly idea emerged beyond its first and innocent form it drew to it a more serious sacrificial idea.’ The sacramental side was subordinated to the sacrificial. In the end it came to this, that, while the New Testament teaching is that Christ offered himself, now the priest offered him. ‘The priest imitates what Christ did’ (Cyprian). It was an absolutely unscriptural change. When Cyprian described the priest as ‘imitating Christ’s sacrifice, and offering a true and full sacrifice to God the Father’, he was the chief culprit in effecting the change from a sacrificium laudis by the Church to a sacrificium propitiatorium by the priest.