Demons were sent running whenever Jesus was around during His earthly ministry. But He wasn't the only One they were powerless against.
Jesus called the twelve apostles together and gave them power and authority over every demon and power and authority to cure diseases. – Luke 9:1, GW
After this, the Lord appointed 70 other disciples to go ahead of him to every city and place that he intended to go. They were to travel in pairs. …The 70 disciples came back very happy. They said, "Lord, even demons obey us when we use the power and authority of your name!" Jesus said to them, "I watched Satan fall from heaven like lightning. I have given you the authority to trample snakes and scorpions and to destroy the enemy's power. Nothing will hurt you. – Luke 10:1,17-19, GW
That's pretty awesome. Jesus gave both His closest apostles and 70 other followers the power to set people free from demonic possession and oppression. So even if Jesus wasn't physically present, many people believed in Jesus just because of the signs and wonders (including casting out demons) that His disciples performed in Jesus' name.
The apostles of Jesus were casting out many evil spirits even after the death and resurrection of Jesus Christ, which only proved His divine nature. They were now continuing the ministry which Jesus had started by preaching the gospel, healing, and helping people get free from demonic possession or oppression.
Now by the hands of the apostles, many signs and wonders took place among the people. …so that they were bringing the sick and placing them on couches and pallets, so that as Peter was passing, at least his shadow might overshadow some of them. And also the multitude from the surrounding cities were coming together into Jerusalem, bringing the sick and those who were tormented by unclean spirits, who were all being healed. – Acts 5:12,15,16, EMTV
The most extraordinary apostle of Jesus Christ was not one of the first disciples. He didn't even believe in Jesus until after He died. Originally, he persecuted the early Christians. I'm talking about the apostle Paul who wrote the majority of the New Testament. He also had the power to cast out evil spirits.
And God was doing extraordinary miracles by the hands of Paul, so that even handkerchiefs or aprons that had touched his skin were carried away to the sick, and their diseases left them and the evil spirits came out of them. – Acts 19:11,12, ESV
One day when we were going to the place of prayer, a female servant met us. She was possessed by an evil spirit that told fortunes. She made a lot of money for her owners by telling fortunes. …Paul became annoyed, turned to the evil spirit, and said, "I command you in the name of Jesus Christ to come out of her!" As Paul said this, the evil spirit left her. – Acts 16:16-18, GW
So, we see that any disciple of Jesus that He appointed had the same power over demons as He did. Specially chosen followers of Christ had authority over evil spirits both during the earthly ministry of Jesus, and after Jesus rose from the dead. So if people missed out on getting set free from demonic possession while Jesus was alive on earth, they still had a chance to be touched by one of His disciples as the early Christian church was being formed after Jesus died for our sins.
But what about today? What if we are oppressed by an evil spirit in the 21st century? These questions will be addressed in the upcoming article: Understanding demons Bible-style, X, today's believers power over demons.
Read this series from the beginning: Understanding demons bible-style, an introduction