Reformed theologians who hold to penal substitutionary atonement emphasize a) the divine nature of Christ and the increased capacity for suffering that that implies and b) the intensity of God's wrath against him.
Louis Berkhof, in his Systematic Theology (3.2.1.B), writes:
[Christ's] capacity for suffering was commensurate with the ideal character of His humanity, with His ethical perfection, and with His sense of righteousness and holiness and veracity. No one could feel the poignancy of pain and grief and moral evil as Jesus could.
[Christ] was subject not only to physical, but also to eternal death, though He bore this intensively and not extensively, when He agonized in the garden and when He cried out on the cross, "My God, my God, why hast thou forsaken me?" In a short period of time He bore the infinite wrath against sin to the very end and came out victoriously. This was possible for Him only because of His exalted nature.
The Heidelberg Catechism, answer 37, reads:
That during his whole life on earth, but especially at the end, Christ sustained
in body and soul the wrath of God against the sin of the whole human race. This he did in order that, by his suffering as the only atoning sacrifice, he might deliver us, body and soul, from eternal condemnation, and gain for us God’s grace, righteousness, and eternal life.
Wayne Grudem, in chapter 27 of his Systematic Theology, emphasizes the meaning of Christ's words on the cross as signifying the completion of his suffering and his victory:
Jesus was able to bear all the wrath of God against our sin and to bear it to the end. No mere man could ever have done this, but by virtue of the union of divine and human natures in himself, Jesus was able to bear all the wrath of God against sin and bear it to the end. [...] When Jesus knew that he had paid the full penalty for our sin, he said, "It is finished" (John 19:30). If Christ had not paid the full penalty, there would still be condemnation left for us. (p578)
Other views of the atonement
Grudem notes that Roman Catholic teaching on the completion of Christ's sacrifice is different. He quotes Ludwig Ott's Fundamentals of Catholic Dogma:
In the Sacrifice of the Mass and in the Sacrifice of the Cross the Sacrificial Gift and the Primary Sacrificing Priest are identical; only the nature and the mode of the offering are different.... according to the Thomistic view, in every Mass Christ also performs an actual immediate sacrificial activity.
Protestants and others who reject the theory of penal substitution also answer this question differently. Proponents of the moral influence theory, for example, do not suggest that Christ was "paying for our sins" at all, and that therefore punishment of eternal length was not required.
Similarly, under the governmental theory, Christ suffers to show that a penalty must be paid for lawlessness, but he does not pay the penalty for all the sins of the elect, so again, full payment is not required. More on these and other theories of the atonement can be found on the corresponding Wikipedia articles, and analysis of them is found in both works of Grudem's and Berkhof's that I've quoted.