It was the power in the name of Jesus plus the disciples' faith in that power that made the demons subject unto them. This can be seen in Acts 19:13-17 where certain Jews called on the name of Jesus in an effort to cast demons out of a man. They used the name of Jesus, but it didn't work because they had not put faith in that name.

Those who relegate demonic spirits to the realm of superstition do not believe in the entire Word of God. The Gospels alone contain over 90 references to the devil or devils (demons). 

The Paul said when writing to these people in Ephesus, "We wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places" (Ephesians 6:12). The devil and demons do exist, but all believers in Jesus share in His total victory over them.

The name of Jesus is not magic. It does not work like a charm. As Peter said, "His name, through faith in his name" (Acts 3:16) is what brings results.