What is the sacramental principle at work with Anointing of the Sick? Of the seven sacraments in the Catholic Church.  I see how each of them are some form of Sacramentum, an oath between man and God. 


Marriage has a marital vow
Holy orders is a vow to be a priest, deacon or bishop.
Reconciliation is a resolution to "Go and sin no more"
In Holy Communion we say, Amen to "The Body of Christ"
In Baptism we resolve to enter more into the Body of Christ
and in Confirmation we renew and confirm that resolution.


But, what is the oath in Anointing of the Sick?  To die peacefully?
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,


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]
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]
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]
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.
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]
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]
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]