From the Papal Magisterium of Ven. Pius XII
From the Encyclical Mystici Corporis :
28. That He completed His work on the gibbet of the Cross is the unanimous teaching of the holy Fathers who assert that the Church was born from the side of our Savior on the Cross like a new Eve, mother of all the living.  "And it is now," says the great St. Ambrose, speaking of the pierced side of Christ, "that it is built, it is now that it is formed, it is now that it is...molded, it is now that it is created... Now it is that arises a spiritual house, a holy priesthood."  One who reverently examines this venerable teaching will easily discover the reasons on which it is based.
29. And first of all, by the death of our Redeemer, the New Testament took the place of the Old Law which had been abolished; then the Law of Christ together with its mysteries, enactments, institutions, and sacred rites was ratified for the whole world in the blood of Jesus Christ. For, while our Divine Savior was preaching in a restricted area - He was not sent but to the sheep that were lost of the House of Israel  - the Law and the Gospel were together in force;  but on the gibbet of His death Jesus made void the Law with its decrees  fastened the handwriting of the Old Testament to the Cross,  establishing the New Testament in His blood shed for the whole human race. "To such an extent, then," says St. Leo the Great, speaking of the Cross of our Lord, "was there effected a transfer from the Law to the Gospel, from the Synagogue to the Church, from the many sacrifices to one Victim, that, as Our Lord expired, that mystical veil which shut off the innermost part of the temple and its sacred secret was rent violently from top to bottom." 
30. On the Cross then the Old Law died, soon to be buried and to be a bearer of death,  in order to give way to the New Testament of which Christ had chosen the Apostles as qualified ministers;  and although He had been constituted the Head of the whole human family in the womb of the Blessed Virgin, it is by the power of the Cross that our Savior exercises fully the office itself of Head of His Church. "For it was through His triumph on the Cross," according to the teaching of the Angelic and Common Doctor, "that He won power and dominion over the gentiles"; by that same victory He increased the immense treasure of graces, which, as He reigns in glory in heaven, He lavishes continually on His mortal members; it was by His blood shed on the Cross that God's anger was averted and that all the heavenly gifts, especially the spiritual graces of the New and Eternal Testament, could then flow from the fountains of our Savior for the salvation of men, of the faithful above all; it was on the tree of the Cross, finally, that He entered into possession of His Church, that is, of all the members of His Mystical Body; for they would not have been untied to this Mystical Body through the waters of Baptism except by the salutary virtue of the Cross, by which they had been already brought under the complete sway of Christ.
31. But if our Savior, by His death, became, in the full and complete sense of the word, the Head of the Church, it was likewise through His blood that the Church was enriched with the fullest communication of the Holy Spirit, through which, from the time when the Son of Man was lifted up and glorified on the Cross by His sufferings, she is divinely illumined. For then, as Augustine notes,  with the rending of the veil of the temple it happened that the dew of the Paraclete's gifts, which heretofore had descended only on the fleece, that is on the people of Israel, fell copiously and abundantly (while the fleece remained dry and deserted) on the whole earth, that is on the Catholic Church, which is confined by no boundaries of race or territory.
29. Ambrose, In Luc, II, 87: Migne, P.L., XV, 1585.
30. Cf. Matth., XV, 24.
31. Cf. St. Thos., I-II, q. 103, a. 3, ad 2.
32. Cf. Eph., II, 15.
33. Cf. Col., II, 14.
34. Cf. Matth., XXVI, 28; I Cor., XI, 25.
35. Leo the Great, Serm., LXVIII, 3: Migne, P.L. LIV, 374.
36. Jerome and Augustine, Epist. CXII, 14 and CXVI, 16: Migne, P.L., XXII, 924 and 943; St. Thos., I-II, q. 103, a. 3, ad 2; a. 4; ad 1; Council of Flor. pro Jacob.: Mansi, XXXI, 1738.
37. Cf. II Cor., III, 6.
38. Cf. St. Thos. III, q. 42, a. 1.
39. Cf. De pecc. orig., XXV, 29: Migne, P.L., XLIV, 400.