To even ask the question seems bizarre but that's a surface response. Whether he likes it or not the Devil serves the sovereign God. There must have been a time when the one we know as Satan made a sinful choice which developed into a habit and became a character and then a life. He is presented in scripture as nothing less than God's tireless enemy. Whatever he does he does with a wicked and anti-God, anti-holiness and anti-life agenda. How then can he be God's servant?
He can be God's servant in the same way a wicked and predatory nation can be God's servant. Brutal Assyria does the will of God without either knowing this to be the case or wanting it to be the case (see Isaiah 10:5-7). Barbaric Babylon is God's scourge of godless Judah in the book of Habakkuk though she is anti-God.
The thorn in Paul's flesh was given to him for a holy purpose (2 Corinthians 12:7) so we know where it came from. Twice he tells us it was to keep him from being arrogant and self-sufficient (note the context of 7-10) so that he might be a better servant of God. That was why he was given the pain so we know it wasn't given by Satan. And yet we're told it was delivered by a satanic messenger, one of Satan's lackeys. It doesn't matter that God's gift came by the hand of a satanic messenger that meant only evil. It was God's gift to Paul that kept Paul from sin and shame and made him a more profoundly useful servant.
Satan was up to his neck in the atoning death of Jesus Christ (since it was his spirit and character that Judas modeled in betraying Christ). He meant no good thing and nor did the Jews and Gentiles who gathered together in heart and mind to murder the Son of God. But in all of this God was fulfilling his own gracious, holy and redeeming purpose. Satan and sinners serve God! See Acts 2:23 and 4:24-28 making that very point.
Satan is God's servant whether he means to be or wants to be! The brothers of Joseph eased their spiteful hearts on Joseph by selling him into Egypt . They had no intention of serving God but God was using them to accomplish what he had already purposed to do. Contrary to the “plain facts” Joseph tells them (Genesis 45:8), “It was not you who sent me here, but God.” He insists that God sent him ahead of them to save lives (45:5, 7 and see 50:20). It isn't a matter of God seeing the eggs were broken and decided to make an omelet. God isn't forever chasing around exercising damage control. Our sinful choices are real choices but God is the Lord of Sin as well as Lord of everything else and Sin and Satan serve his purposes.
To save the soul of an ungodly man Paul said he was to be given over to Satan (1 Corinthians 5:5) and two men were given to Satan so that they'd learn not to blaspheme (1 Timothy 1:20 ). Those two passages beg to be developed but however that's to be done they make the handing over to Satan a part of their redemption. Since the handing over is the will of God and the purpose of it is their redemption we have Satan serving God's purpose!
In the book of Job (chapters 1 & 2) we have Satan acting on God's behalf in putting Job to grief. In 1:11 and 2:5 Satan challenges God to touch Job and put him to the test. God sends Satan to carry out that job for him. Satan does not do this of his own power or of his own accord! It is God's fire that burns Job's crops (1:16 ) and it is Yahweh that brings trouble on Job (42:11). And Job knows it (1:20-22 and 2:10 ). God puts Job to grief for a redemptive purpose and uses spiteful, cynical Satan as his message-boy to get the job done. And compare 2 Samuel 24:1 and 1 Chronicles 21:1. These two passages leave us to choose whether God used Satan or Satan used God. The answer seems clear to me.