As Disciples of Christ they invoked HIS name, to whom their souls belong. The Bible says whomever belongs to Christ (this is not an exact quote, but there are many here who can offer it) can invoke his name in casting out demons and it shall be done. So in a sense any believer can perform an exorcism. The Catholic Church attaches a lot of ritual to the act, but Jesus used to just say GET OUT!! and the demon left the body. I've had the occasion to order a demonic presence from a home and it left to be replaced instantly by the literal warmth of the Holy Spirit. Who am I? Just a sinner saved by Grace. Nothing more.
There's a useful illustration in Hollyweird. In the first Fright Night movie, the phony "Vampire Slayer" faces a real vampire and shakily waves a crucifix in his face and the vampire just laughs at him and crushes the cross with his bony claws. He cackles: Silly man...you've GOT to BELIEVE for that to work... Then along comes the teen aged hero and does the same thing and the vampire shrinks back hissing....Okay, so it's hokey...but true.
But yes, exorcism is one of the jobs that baptised laity can do. Best to get training first though: apparently the chief danger in exorcism (as in healing and other such acts) is the danger of spiritual pride.
Look at this in Mark 3:11-12
"11": And unclean spirits, when they saw him, fell down before him, and cried, saying, Thou art the Son of God.
"12": And he straitly charged them that they should not make him known.
And then there is this about who is doing the casting out of spirits. And keep in mind there are only two spiritual powers...there is Good (God/Yeshua/the Holy Spirit) and there is evil (Satan). If someone doesn't serve God, in their rebellion they serve Satan.
From Mark 3:23-26
23: And he called them unto him, and said unto them in parables, How can Satan cast out Satan?
24: And if a kingdom be divided against itself, that kingdom cannot stand.
25: And if a house be divided against itself, that house cannot stand.
26: And if Satan rise up against himself, and be divided, he cannot stand, but hath an end.