Father God is the one who does the miracles, He is the source of all we ask in Jesus' name and the Holy Spirit brings them into fruition in our natural world, from the heavenly places.
There is so much proof of the power and the ministry of the gifts of the Holy Spirit in both the Old and New Testaments. In 2 Kings chapter four, God used Elisha to multiply the oil for the prophet's widow so she could pay her debt. Then God gave the woman and her elderly husband, a much-desired son to the and later when the child died, God answered Elisha's prayer and brought him back to life. We then read how Elisha turned a poisonous stew into good, nourishing food; the next miracle that we read about in the same chapter was the multiplication of the “twenty loaves of barley bread, and newly ripened grain in a knapsack” that fed one hundred hungry men, as the Lord had spoken that “‘They shall eat and have some left over.’” Then in 2 Kings 5:8-15, we read about the unusual way that God used Elisha to heal the commander of the Syrian army, Naaman, of leprosy.
God has always shown His love for His elect and did miracles for them. In Old Testament times, the Lord would have the Holy Spirit descend upon the prophets when He chose to speak or do miracles. God's people had to go to the prophets or priests of God or to the Tabernacle and later to the Temple of God, when they needed God's help.
Today, we are so blessed to have access to God's presence, at any time, once we receive the Holy Spirit (you may read more about it here: The Baptism of the Holy Spirit).
Jesus Christ came to be a human, as one of us, and lived a perfect, sinless life for God's glory and pleasure and to show us how to live. He set the perfect example for us, as we see it written in the Bible and only with the help of the Holy Spirit can we do it. The life that is available through Jesus is the most abundant life and it cannot compared to any other way of living (John 10:10). Jesus was sent by the Father and willingly gave up His godly power and attributes to live in a human body. With the Help of the Holy Spirit, Jesus performed signs and miracles, healed the sick, cleansed the lepers and resurrected the dead. Then, after Jesus sacrificed His life for us, paying for our sins on the cross, Father God resurrected Him, through the work and power of the same Holy Spirit. All that Jesus did was perfect and we are shown how to follow His example in the following verses.
John 14:12-18 ““Most assuredly, I say to you, he who believes in Me, the works that I do he will do also; and greater works than these he will do, because I go to My Father. And whatever you ask in My name, that I will do, that the Father may be glorified in the Son... If you love Me, keep My commandments. And I will pray the Father, and He will give you another Helper,that He may abide with you forever— the Spirit of truth, whom the world cannot receive, because it neither sees Him nor knows Him; but you know Him, for He dwells with you and will be in you. I will not leave you orphans; I will come to you.””
Jesus is saying that there are some very specific requirements that we must meet in order to receive the Holy Spirit; first we have to believe that Jesus is the Christ and then to love Him by obeying His commandments. By doing so we can accomplish His work and even greater works than Jesus did. We see what the Holy Spirit did through the apostles Peter and Paul as we read throughout the book of Acts
The gifts and ministries of the Holy Spirit did not cease working through God's children when the lives of the early apostles ended. I believe we have all heard of the marvelous miracles that God has done all throughout church history and He continues to do even today.
Acts 5:12, 15 “And through the hands of the apostles many signs and wonders were done among the people.... so that they brought the sick out into the streets and laid them on beds and couches, that at least the shadow of Peter passing by might fall on some of them.”
Acts 19:11-12 “Now God worked unusual miracles by the hands of Paul, so that even handkerchiefs or aprons were brought from his body to the sick, and the diseases left them and the evil spirits went out of them.”
To get to where we may do greater miracles than Jesus did, we need to receive the gifts that the Holy Spirit decides to give us, and then use them for the benefit of those around us, or sometimes even for our own.
You may read more about this in the article “The Gifts and Ministry of the Holy Spirit”