2019-03-29: An updated version of this article is available here: Does the Devil (or Satan) Exist? Friedrich Schleiermacher debates John Calvin

Illustration of the Devil in the Codex Gigas, Early 13th Century

John Calvin's proof for the existence of Satan in Institutes 1.14.17-18 continues to be useful, so I'm providing an extended quotation of the sections where his proof is easily summarized. Too often, people have a Manichaeistic or Dualistic view of God and Satan, as if there are really two gods: a good god and a bad god in constant battle for human souls. I call this a "High View of Satan" where the Devil is so powerful that his power of darkness is always on the verge of overcoming the good god of light. The Judeo-Christian God is an almighty sovereign God, and all things are under his control and all things are planned and happen according to his decretum absolutum (Absolute Decree). Satan therefore is only a tool that God has created for bringing about God chief and final end of glorifying Himself forever. Satan is a servant of God, and truly is the Advesary, but at the same time, all of his actions that are intended for evil will ultimately serve God's glorious ends to his own chagrin and destruction.

Calvin's proof derives from Theodicy (the problem of evil.) How may a good God be sovereign over all things, including evil, without being himself evil. If all things happen according to God's will, then evil must be God's will? It's from this dilemma that Calvin demonstrates how Satan has a necessary place due in the Theatre of God's Glory.  

Institutes I.14.17 With regard to the strife and war which Satan is said to wage with God, it must be understood with this qualification, that Satan cannot possibly do anything against the will and consent of God. For we read in the history of Job, that Satan appears in the presence of God to receive his commands, and dares not proceed to execute any enterprise until he is authorised. In the same way, when Ahab was to be deceived, he undertook to be a lying spirit in the mouth of all the prophets; and on being commissioned by the Lord, proceeds to do so. For this reason, also, the spirit which tormented Saul is said to be an evil spirit from the Lord, because he was, as it were, the scourge by which the misdeeds of the wicked king were punished. In another place it is said that the plagues of Egypt were inflicted by God through the instrumentality of wicked angels. In conformity with these particular examples, Paul declares generally that unbelievers are blinded by God, though he had previously described it as the doing of Satan. [Job 1:6; Job 2:1; 1 Kings 22:20; 1 Sam 16:14; 1 Sam 18:10; 2 Thess 2:9; 2 Thess 2:11.] It is evident, therefore, that Satan is under the power of God, and is so ruled by his authority, that he must yield obedience to it. Moreover, though we say that Satan resists God, and does works at variance with His works, we at the same time maintain that this contrariety and opposition depend on the permission of God. I now speak not of Satan’s will and endeavour, but only of the result. For the disposition of the devil being wicked, he has no inclination whatever to obey the divine will, but, on the contrary, is wholly bent on contumacy and rebellion. This much, therefore, he has of himself, and his own iniquity, that he eagerly, and of set purpose, opposes God, aiming at those things which he deems most contrary to the will of God. But as God holds him bound and fettered by the curb of his power, he executes those things only for which permission has been given him, and thus, however unwilling, obeys his Creator, being forced, whenever he is required, to do Him service.

Institutes I.14.18 God thus turning the unclean spirits hither and thither at his pleasure, employs them in exercising believers by warring against them, assailing them with wiles, urging them with solicitations, pressing close upon them, disturbing, alarming, and occasionally wounding, but never conquering or oppressing them; whereas they hold the wicked in thraldom, exercise dominion over their minds and bodies, and employ them as bond-slaves in all kinds of iniquity. Because believers are disturbed by such enemies, they are addressed in such exhortations as these: “Neither give place to the devil;” “Your adversary the devil, as a roaring lion, walketh about seeking whom he may devour; whom resist steadfast in the faith,” (Eph. 4:27; 1 Pet. 5:8). Paul acknowledges that he was not exempt from this species of contest when he says, that for the purpose of subduing his pride, a messenger of Satan was sent to buffet him (2 Cor. 12:7). This trial, therefore, is common to all the children of God. But as the promise of bruising Satan’s head (Gen. 3:15) applies alike to Christ and to all his members, I deny that believers can ever be oppressed or vanquished by him. They are often, indeed, thrown into alarm, but never so thoroughly as not to recover themselves. They fall by the violence of the blows, but they get up again; they are wounded, but not mortally. In fine, they labour on through the whole course of their lives, so as ultimately to gain the victory, though they meet with occasional defeats. We know how David, through the just anger of God, was left for a time to Satan, and by his instigation numbered the people (2 Sam. 24:1); nor without cause does Paul hold out a hope of pardon in case any should have become ensnared by the wiles of the devil (2 Tim. 2:26). Accordingly, he elsewhere shows that the promise above quoted commences in this life where the struggle is carried on, and that it is completed after the struggle is ended. His words are, “The God of peace shall bruise Satan under your feet shortly,” (Rom. 16:20). In our Head, indeed, this victory was always perfect, because the prince of the world “had nothing” in him (John 14:30); but in us, who are his members, it is now partially obtained, and will be perfected when we shall have put off our mortal flesh, through which we are liable to infirmity, and shall have been filled with the energy of the Holy Spirit. In this way, when the kingdom of Christ is raised up and established, that of Satan falls, as our Lord himself expresses it, “I beheld Satan as lightning fall from heaven,” (Luke 10:18). By these words, he confirmed the report which the apostles gave of the efficacy of their preaching. In like manner he says, “When a strong man armed keepeth his palace, his goods are in peace. But when a stronger than he shall come upon him, and overcome him, he taketh from him all his armour wherein he trusted, and divideth his spoils,” (Luke 11:21, 22). And to this end, Christ, by dying, overcame Satan, who had the power of death (Heb. 2:14), and triumphed over all his hosts that they might not injure the Church, which otherwise would suffer from them every moment. For (such being our weakness, and such his raging fury), how could we withstand his manifold and unintermitted assaults for any period, however short, if we did not trust to the victory of our leader? God, therefore, does not allow Satan to have dominion over the souls of believers, but only gives over to his sway the impious and unbelieving, whom he deigns not to number among his flock. For the devil is said to have undisputed possession of this world until he is dispossessed by Christ. In like manner, he is said to blind all who do not believe the Gospel, and to do his own work in the children of disobedience. And justly; for all the wicked are vessels of wrath, and, accordingly, to whom should they be subjected but to the minister of the divine vengeance? In fine, they are said to be of their father the devil. (2 Cor 4:4; Eph 2:2; Rom 9:22; John 8:44; 1 John 3:8) For as believers are recognised to be the sons of God by bearing his image, so the wicked are properly regarded as the children of Satan, from having degenerated into his image.

John Calvin, Institutes of the Christian Religion, 1.14.17-18, beveridge translation, http://www.ccel.org/ccel/calvin/institutes.iii.xv.html#iii.xv-Page_153