According to the Catholic Encyclopedia article on Exorcism:

"Assuming the reality of demonic possession, for which the authority of Christ is pledged, it is to be observed that Jesus appealed to His power over demons as one of the recognised signs of Messiahship (Matthew 12:23,12:28; Luke 11:20). He cast out demons, He declared, by the finger or spirit of God, not, as His adversaries alleged, by collusion with the prince of demons (Matthew 12:24,12:27; Mark 3:22; Luke 11:15,11:19); and that He exercised no mere delegated power, but a personal authority that was properly His own, is clear from the direct and imperative way in which He commands the demon to depart (Mark 9:24; cf. 1:25 etc.): "He cast out the spirits with his word, and he healed all that were sick" (Matthew 8:16). Sometimes, as with the daughter of the Canaanean woman, the exorcism took place from a distance (Matthew 15:22 sqq.; Mark 7:25). Sometimes again the spirits expelled were allowed to express their recognition of Jesus as "the Holy One of God" (Mark 1:24) and to complain that He had come to torment them "before the time", i.e the time of their punishment (Matthew 8:29 sqq; Luke 8:28 sqq.). If demoniac possession was generally accompanied by some disease, yet the two were not confounded by Christ, or the Evangelists. In Luke 13:32, for example, the Master Himself expressly distinguishes between the expulsion of evil spirits and the curing of disease. Christ also empowered the Apostles and Disciples to cast out demons in His name while He Himself was still on earth (Matthew 10:1,Matthew 10:8; Mark 6:7; Luke 9:110:17), and to believers generally He promised the same power (Mark 16:17). But the efficacy of this delegated power was conditional, as we see from the fact that the Apostles themselves were not always successful in their exorcisms: certain kinds of spirits, as Christ explained, could only be cast out by prayer and fasting (Matthew 17:15,17:20; Mark 9:27-28; Luke 9:40). In other words the success of exorcism by Christians, in Christ's name, is subject to the same general conditions on which both the efficacy of prayer and the use of charismatic power depend. Yet conspicuous success was promised (Mark 16:17). St. Paul (Acts 16:18,19:12), and, no doubt, the other Apostles and Disciples, made use of regularly, as occasion arose, of their exorcising power, and the Church has continued to do so uninterruptedly to the present day. "