Here are some possibilities:
1. He asked the Father, and the Father did them.
2. He was God in the flesh, so He did them as God.
3. He was filled with the Spirit and the power of the Spirit was on Him as He did them, much as the Apostles later did them.
4. All or a combination of one or more of the three options above.
Here's one argument:
First,we know that, “although [Jesus] existed in the form of God, … [He] emptied Himself,taking the form of a bond-servant, and being made in the likeness of men.”Philippians 2:6-7.Indeed, He “did notregard equality with God a thing to be grasped.” Philippians 2:6.
Second, there is virtually no evidencethat the Father did anything directly and not by the Holy Spirit at the requestof Jesus.
The closest text says:“But if I cast out demonsby the finger of God,then the kingdom of God has comeupon you.”Luke 11:20.However, Matthew 12:28says it even more specifically:“Butif I cast out demonsby the Spirit of God, then the kingdom ofGod has come upon you.”
He also says:
The Spirit of the Lord is upon Me, Because Heanointed Me to preach the gospel to the poor. He has sent Me to proclaim release to the captives, Andrecovery of sight to the blind, Toset free those who are oppressed,Luke 4:18.Not only do Matthew12:28 and Luke 4:18 settle the matter as to exorcisms (freeing the oppressed)and healings (sight to the blind), but they help explain that God did certainthings in the Old Testament through the Holy Spirit, though they were said tobe done by the “finger of God.” (Cf., Plagueof gnats, Exodus 8:19; and the Ten Commandments written in stone, Exodus 31:18;Deuteronomy 9:10).
Virtuallyall of Jesus’ healings, raisings from the dead and exorcisms were performed bythe power of the Holy Spirit:
You know ofJesus of Nazareth, how God anointedHim with the Holy Spiritand with power, and how He went about doing good and healing all who were oppressed by the devil,for God was with Him.Acts 10:38.So far as His miraclesof changing water to wine, walking on water and changing the weather,multiplying loaves and fish, catching many fish, putting coins in their mouths and beingtransfigured, there is no definitive proof.However, between the definition of “the finger of God” and the workingof the Spirit in healings and exorcisms, there is no real reason to suspectanother agent besides the Holy Spirit.
Theclosest thing to an ambiguity is where Jesus tells the disciples the reason whythey could not cast out the more difficult demon.He says, “This kind cannot come out byanything but prayer.”Mark 9:29.So, does that mean the Father must directlyintervene because the Spirit is unable?Or, far more likely, is it that the authority and/or power necessarywith such demons must be increased via an answer to prayer?
Sofar as the authoritative sermons of Jesus, He Himself testifies:“He whom God has sent speaks the words ofGod; for He gives the Spirit without measure.”John 3:34.
So far as knowing the thoughts of a person,Jesus seems to directly apprehend these without any intervening petition to theFather, indicating that, if this was miraculous, it was through the power ofthe Spirit rather than through petition to the Father:
Immediately Jesus, aware in His spiritthat they were reasoning that way within themselves, said to them, “Why are youreasoning about these things in your hearts?Mark 2:8
Finally, the fact that Jesus was ableto give the disciples authority and power to heal and cast out demons (Matthew10:1) is consistent with how the Spirit has functioned in the past.Elijah gave a double portion of the Spiritwhich was on Him to Elisha.2 Kings2:9-25.Indeed, if we examine the typesof miracles performed by Elijah and Elisha in 1st & 2ndKings, we see that by the power of the Spirit upon them they performed manymiracles with water, weather, multiplying of food and many others, not tomention healings and even raising from the dead.
Edited point: I made this same argument here, in a thread with a similar title, but it was never addressed by anyone.