What is the sacramental principle at work with Anointing of the Sick?

The Catechism of the Catholic Church says that "the first grace of this Sacrament is one of strengthening, peace and courage to overcome the difficulties that go with the condition of serious illness or the frailty of old age." There's quite a bit more: it covers six pages. "In this sacrament the sick person receives the strength and the gift of uniting himself more closely to Christ's Passion: in a certain way he is consecrated to bear fruit by configuration to the Saviour's redemptive Passion."

So one might say, 7: In Anointing, we recognise our own Cross.

However, I'm not sure that an oath can be brought out of that in the same way as your other six interpretations. I believe this sacrament is one of pure grace.

I'll have another go since Affable Geek has redefined the question...

The noted Anglican divine Richard Hooker defined a sacrament as "an outward and visible sign of an inward and spiritual grace" [sorry, don't have a reference for that]. The Roman Catholic Church teaches "the sacraments are efficacious signs of grace, instituted by Christ and entrusted to the Church, by which divine life is dispensed to us. The visible rites by which the sacraments are celebrated signify and make present the graces proper to each sacrament. They bear fruit in those who receive them with the required dispositions." [Catechism of the Catholic Church, s.1131]

The sacraments are a means of grace. In fact sacramentum means "made holy". An oath makes a promise holy (and so may be translated as sacramentum); but a sacrament in the Christian sense, although holy, is not a promise or oath. This is why finding a promise or oath in Anointing is so difficult: it is purely a means of grace.

I'm not qualified to say whether it's right to discern an oath/promise in the other sacraments, but I suspect that what Peter discerns as a promise is in fact a manifestation of the grace of the sacrament.

Taking the seven sacraments in Peter's order,

1. Marriage — the grace of divine love. The sacrament of Matrimony signifies the union of Christ and the Church. It gives spouses the grace to love each other with the love with which Christ has loved his Church; the grace of the sacrament thus perfects the human love of the spouses, strengthens their indissoluble unity, and sanctifies them on the way to eternal life (cf Council of Trent: DS 1799). [CCC 1661]

2. Orders — the grace of service in the Church. The ministerial priesthood differs in essence from the common priesthood of the faithful because it confers a sacred power for the service of the faithful. The ordained ministers exercise their service for the People of God by teaching (munus docendi), divine worship (munus liturgicum) and pastoral governance (munus regendi). [CCC 1592]

3. Reconciliation — the grace of forgiveness and reconciliation with God and the Church. The spiritual effects of the sacrament of Penance are: reconciliation with God by which the penitent recovers grace; reconciliation with the Church; remission of the eternal punishment incurred by mortal sins; remission, at least in part, of temporal punishments resulting from sin; peace and serenity of conscience, and spiritual consolation; an increase of spiritual strength for the Christian battle. [CCC 1496]

4. Communion — the Real Presence of Christ in the Sacrament. By the consecration the transubstantiation of the bread and wine into the Body and Blood of Christ is brought about. Under the consecrated species of bread and wine Christ himself, living and glorious, is present in a true, real, and substantial manner: his Body and his Blood, with his soul and his divinity (cf. Council of Trent: DS 1640; 1651). [CCC 1413] Communion with the Body and Blood of Christ increases the communicant's union with the Lord, forgives his venial sins, and preserves him from grave sins. Since receiving this sacrament strengthens the bonds of charity between the communicant and Christ, it also reinforces the unity of the Church as the Mystical Body of Christ. [CCC 1416] Transubstantiation is quite successfully described in the accepted answer here.

5. Baptism — the grace of forgiveness of sin. The fruit of Baptism, or baptismal grace, is a rich reality that includes forgiveness of original sin and all personal sins, birth into the new life by which man becomes an adoptive son of the Father, a member of Christ and a temple of the Holy Spirit. By this very fact the person baptized is incorporated into the Church, the Body of Christ, and made a sharer in the priesthood of Christ. Baptism imprints on the soul an indelible spiritual sign, the character, which consecrates the baptized person for Christian worship. [CCC 1279-80]

6. Confirmation — the grace of witnessing to the Faith. Confirmation perfects Baptismal grace; it is the sacrament which gives the Holy Spirit in order to root us more deeply in the divine filiation, incorporate us more firmly into Christ, strengthen our bond with the Church, associate us more closely with her mission, and help us bear witness to the Christian faith in words accompanied by deeds. Confirmation, like Baptism, imprints a spiritual mark or indelible character on the Christian's soul. [CCC 1316-17]

7. Unction — the grace of strengthening, peace and courage. The special grace of the sacrament of the Anointing of the Sick has as its effects: the uniting of the sick person to the passion of Christ, for his own good and that of the whole Church; the strengthening, peace, and courage to endure in a Christian manner the sufferings of illness or old age; the forgiveness of sins, if the sick person was not able to obtain it through the sacrament of Penance; the restoration of health, if it is conducive to the salvation of his soul; the preparation for passing over to eternal life. [CCC 1532]